Mathematical models preview book by Δημήτρης Αντωνίου

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Δημήτρης Αντωνίου PhD(Hon), FRCS,Γενικός Χειρουργός, Ογκολόγος

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Απαγορεύεται η

αναδημοσίευση και γενικά η αναπαραγωγή του παρόντος έργου (πλήρους

έκδοσης) με οποιοδήποτε τρόπο, τμηματικά ή περιληπτικά, στο πρωτότυπο ή σε μετάφραση, ή άλλη διασκευή, χωρίς γραπτή άδεια του εκδότη και συγγραφέα, σύμφωνα με τον ν. 2121/1993 και την Διεθνή Σύμβαση της Βέρνης (που κυρώθηκε με τον ν. 100/1975).

CONTENTS CONTENTS 1. PRINCIPLES OF PHILOSOPHY OF SCIENCE AND MATHEMATICS

2. CHAPTER-I a. DEFINITIONS

b. EXPONENTIAL GROWTH MODELS

c. LOGISTIC GROWTH MODELS (Verhulst)

d. GOMPERTZ GROWTH MODELS

3. CHAPTER-II a. FRACTAL GROWTH AND POWER LAWS

4. CHAPTER-IIII a. DIFFUSION GROWTH MODELS

103

5. CHAPTER-IV MATHEMATICAL MODELS OF TUMOUR-IMMUNE INTERACTIONS a. I.A. PREDATOR-PREY GROWTH SYSTEMS (Lotka-Volterra equations)

165

b. I.B. PREDATOR-PREY MODELS OF TUMOURIMMUNE INTERACTIONS

172

c. II.A. LOTKA-VOLTERRA COMBINED WITH LOGISTIC GROWTH SYSTEMS

174

d. II.B. LOTKA-VOLTERRA COMBINED WITH LOGISTIC MODELS OF TUMOUR-TUMOUR INTERACTIONS

175

6. APPENDIX-I,II,III

177, 257, 303

7. BIBLIOGRAPHY-I,II,III

245 ,295,377

PRINCIPLES OF PHILOSOPHY OF SCIENCE PRINCIPLES OF PHILOSOPHY OF SCIENCE AND MATHEMATICS Scientific knowledge consists of scientific beliefs and it is an exclusive human mental essential feature, which provides the human with an evolutionary advantage to dominate over the physical and to make itself the leading, crucial and determining factor of the natural selection and evolution and finally to make itself able to determine feely its own fate and destiny. Therefore Science seeks to see ―behind the physical veil and curtains‖, i.e. to reveal the unobserved behind the physical appearances (phenomena) and use it for achieving the above goals. The mere experience of the physical world does not constitute epistemic knowledge. Scientific beliefs are justified (rational) statements for the truth or falsity of a special kind of unobservable metaphysical abstract entities which are ―assertoric‖ and which hereafter will be called entities S. ―Assertoric‖ are the unobservables (theoretical terms) which ―refer‖ to an abstract entity (existence) which does or might (or could) correspond to, or assert something about the actual physical world, such f.e. ―God‖, ―beauty‖, ―number‖, ―physical law‖ etc. The assertoric abstract statements could be alternatively interpreted as ―potentially corresponding complex‖ to the actual world and they could be either true or false. Which is the criterion of them being true or false? The answer is that the criterion is as to whether they do actual correspond or not to the things as the latter stand in the world. If the assertoric terms correspond to the things in the world, they are true, if they do not, they are false. This contrasts to the unobservable non-assertoric and non-corresponding non-entities such as f.e. ―welcome‖, which express only an attitude, or ―zero‖, ―emptiness‖, ―non-existence‖, ―hallucinations‖, ―flying elephant‖ etc., which do not refer to any abstract entity which does or could correspond to, or assert something about the actual world whatsoever. The non-assertoric statements are neither true nor false and exist outside of our cognitive epistemic domain (they are unknowable). Equivalently we can say that scientific beliefs are the justifying consciousness of the truth or falsity of the S. We, at this point, are commending on the ontology of the entities S, which is the differentiating criterion of the two major poles of ontology in philosophy, those of Realism and Antirealism. The latter is divided in the ―pure empiricism‖ (Hume, Berkeley, Locke), logical empiricism (Carnap, Quine, Reichenbach), constructive empiricism (van Fraassen) and social constructivism (Kuhn, Feyerabend, late Wittgenstein). We restrict our discussion mainly on Realism and social constructivism type of antirealism because these are the most important ontologies in the philosophy of science. Realism (Plato, Hegel, Spinoza) claims that

DIMITRIOS ANTONIOU Mental (social) Constructivism (MC) (by Th. Kuhn 1962, Wittgenstein) Social (mental) Constructivism claims that the pessimistic metainduction problem can be solved by exclusively assigning to each rival and empirical equivalent theory one particular definite ―world‖, the latter being heavily ―theory-laden‖ contaminated and observational, and by rejecting from its domain all the parts responsible for scepticism in Realism. MC claims that theories are mental constructs (anti-realism), that our experience consists of theory-laden observations, that is mental interpretations of our experience, making thus the correspondence principle and any kind of formal logic (including the classical Aristotelian logic) used in the latter, totally unnecessary as a criterion of theory truth. Theory-laden theory in MC introduces an anti-realistic phenomenalistic view of the world. An example of a theory-laden observation is the next picture, in which we see either a rabbit, or a duck, depending on our mental intension (or idea) to see either the one, or the other animal. The «Rabbit-Duck» of L. Wittgenstein

According to the above, a mental constructivist would claim that there are not true mathematical theories beyond those that are provable by the mathematician minds. Based on these two original principles MC goes on to solve the sceptical problem of many rival and empirical equivalent theories, by constructing for each rival theory a separate and independent ―total‖ world (holism), which can ―exist‖ in parallel, or in succession, with its rival worlds (theories). The rationality (―logic‖) used for the creation of each theory-world –is not any more any kind of formal logic- but instead its “coherence” (see below), which implies that MC rejects the ―prove‖ by contradiction and it does not consider the ―law of excluded middle‖ as ―axiomatic law‖ and this because ―coherence‖ cannot coexist with the ―law of non-

PRINCIPLES OF PHILOSOPHY OF SCIENCE contradiction‖. We have already constructed a the world of General Relativity based upon Lobachevsky`s logic which by neglecting the ―law of excluded middle‖ ―constructs‖ Riemann`s geometry of General Relativity. The connecting ―back-bone‖ of each particular theory-world consists of its pluralistic uses of its objects under the authority of only one ―physical ―law-rule‖, that of ―Coherence‖, which demands that all uses of every particular object should be coherent with each other and with all uses of all other objects in the world. This, at this point, leads to a holistic view of the world, but as we shall see latter this is not necessarily always the case. Coherently with the above, MC modifies the essential meaning of the term ―knowledge‖, as it is conceived in realism, namely as JTBs, and it instead advocates that ―knowledge‖ means the agent ―to be able to decide the possible uses of the things in the world‖, which equals to ―knowing the meaning of the thing down in earth‖. Or, equivalently, that the agent ―can decide how to apply the rules (laws) under different circumstances‖. The coherent use of the things in each theory-world, can result inductively in generating elementary formal ―physicalmathematical laws‖, which could give rise to theorems (theories) with physical content. The elementary formal ―physical-mathematical laws‖ should be conceived as the Syntax of my language by which I construct the content of my language by which I describe and understand, i.e. to know the meaning) of my world. The Syntax (Form) of my world is not itself transcendental and mind-independent, unknowable and meaningless (as it is in structural realism), but it is totally knowable, because it is a construction of the mind, coevolved and coexistent along with the intensional use of thing by the latter. The Syntax could be either the grammatical rules of my everyday common language, or of my content mathematical language )‖physical laws‖ describing formally my world. ―My language is my world‖ (Wittgenstein). Between Syntax (Form) and Content (meaning) of the World, there exist a relation of ―intensionality‖ (Husserl), or of ―Existentialism‖ (Sartre, Heidegger, Wittgenstein). The meaningful objects exist in the world as ―values of human action (use) on them‖, they are depositions of human action and use. But what really means ―to know, or understand the formalistic Syntax‖? ―Knowing the Syntax‖ (the rules), means that I am capable of deciding how to apply the Syntax (rules) under different circumstances. We therefore do not interpret the Syntax, but we instead make reliable decisions for its applications based on actual-pragmatic facts of the world. Different circumstances imply different Syntax applications. Thus the resulted meaning is “contextual” and “pragmatic”. Syntax and Content (meaning) are related with a mutual self-justification, self-modification and dialectic confirmation (not in the strict Hegelian sense).

DIMITRIOS ANTONIOU biological patterns based of lateral inhibition and different rates of diffusion for the participants and cell pattern-preserving growth models with polymorphism of the individual cells, are isomorphic too. Anti-Scepticism, through Isomorphism: A Realist`s response We remind: Isomorphism claims structure invariance across theory`s content change. The feature of Structural Realism (SR), which differentiates it crucially from Typical Realism, is that SR refers and evaluates theories in respect to their “abstract, mathematical form”, rather to their physical contents. The significance and implications of this difference will be immediately justified. The structure can be instantiated in the world we perceive in two different ways, which also determine the anti-sceptic mode of action of Structural Realism: i) the structure is instantiated on the things we perceive over and above the physical content of the theories which populate it, like f.e. we recognize the same melody (structure) being played in different keys (contents). This kind of instantiation argues against the empirical equivalence form of scepticism. Explicitly: The application of isomorphism to the empirical equivalence type of scepticism, implies the meaningless of the latter (or its non-existence at all), because the incompatibility of the (phenomenally) empirical equivalent theories are attributed to their different contents, whereas their basic structure is maintained invariant under theory-change. This implies the rejection of the ―one over many” form of scepticism too (conjoined terms), which nevertheless can be tested and independently, ii) or the structure can be instantiated by determining the pattern in which the objects are manifested in the world, but it does not offer the objects themselves. This kind of instantiation argues against the “one over many” form of scepticism, and refers to the phenomenon of “multiply realizability” (see above). From the above it comes that SR saves supervenence from its sceptical threat encountered in the typical realism`s domain, iii) we said above that the structure can be instantiated by determining the pattern in which the objects are manifested in the world, but it does not offer the objects themselves. This means that SR cannot deliver explanations of the truth conditions of the theories (mathematical statements), since it doesn`t offer us any objects to serve as the referants of the theories. This constitutes another strong response against scepticism, because rejects the necessity of the correspondence principle, and frees the truth seeking process from its dependence on the evidence, which could nevertheless be interpretations of our experience. This further implies that in SR the

DIMITRIOS ANTONIOU The g(N) expresses the "per capita growth of the tumour-Yield", which, in the autonomous systems (exponential and logistic), is independent from the time clock or the commencing time, as shown in the next figures:

Fig 2 a & b. Exponential, Logistic and Gompertzian curves. where K= "carrying capacity" and r = â&#x20AC;&#x2022;actual growth factorâ&#x20AC;&#x2013;.

DIMITRIOS ANTONIOU Sigmoid curves: Why we need to define a term such as the g(N) in our study of these growth models?. This is necessary, because the Logistic and Gompertzian models â&#x20AC;&#x201C;which both two are Sigmoid curves- cannot be distinguished by plotting their N against time-t, or dN/dt against timet, because all the latter are indistinguishable in them both, as it is evident from the following figures.

Fig.1a. Sigmoid curves: Logistic and Gompertzian functions, where: K=carrying capacity, r = actual growth rate.

DEFINITIONS This is aggravated by the fact that in the N(t) curves the inflection points of both two systems (Logistic and Gompertzian) can be identical, provided that the inflection point of the Logistic curve is at the point=K/2, whereas that of the Gompertzian curve is at any point below K/2 (<K/2), where K=carrying capacity is the maximum number of cells that can be reached with the available nutrients. as shown in the next figure: .

Fig.1b. The inflection point of the Logistic curve is at K/s, whereas the inflection point of the Gompertz curve is below K/2 (<K/2). We also see that the humped logistic curve is â&#x20AC;&#x2022;symmetricâ&#x20AC;&#x2013; around the midpoint (=K/2), whereas the Gompertzian curve is not symmetric in that sense. The above figures depicts the a function N(t) in time, i.e. of the number N of population members (tumour cells) in respect to time of the two of the examined models, namely the Logistic and Gomperztian. We see that their curves N(t) are sigmoid, with a definite (and potentially identical) reversing point at K/2. Both curves tells us that: i) the overall N(t) growth is bounded, ii) that below the reversing (inflection) point the dN/dt is increasing (positive) positive, whereas - as the N is still increasing above the reversing point- the dN/dt is decreasing (negative). Therefore the graphical depiction of N(t) or dN/dt in respect to time-t, does not give us any qualitatively significant data by which we could distinguish the two systems in question. On contrary, the distinction between the two sigmoid curves can be achieved: i) by plotting g(N) against N, as shown above in p.34 and in p.61-62. In particular, in the continuous autonomous

EXPONENTIAL GROWTH MODELS EXPONENTIAL CONTINUOUS AND AUTONOMOUS MODEL We define as "growth rate" of the tumour under ideal conditions of unlimited food supply (see ―definitions‖) the differential equation:

In the exponential model the g(N) is constant as a function of N (in relation to N), therefore we can write: g(N) = k = constant. Therefore we take:

(1) where k=g(N)= dK/dt (reproductive constant , or rate of reproduction). The differential equation dN/dt=kN, is ―a first order‖ linear ordinary, ―power law‖, homogenous and autonomous (it does not depend on the variable -t) differential equation. The above function (1) represents the Exponential Model which is the solution of the differential equation dN/dt = kN and which was introduced by the British economist Thomas Malthus 1766-1834.

LOGISTIC GROWTH MODELS LOGISTIC MODEL (Verhulst 1838) A) THE CONTINUOUS AND AUTONOMOUS SYSTEM We think that for the better understanding of the logistic tumour growth model, it is necessary first to describe the application of this model to population dynamics, which it was developed at first hand. This model is more realistic than the exponential, because the former assumes a limited nutrient source availability to animals (or tumour cells, as really is the case due to impairment of the blood supply to the innermost situated cells, caused by the compression of their blood vessels by the tumour itself) and therefore it assumes an intra-species competition for food. This first results in the death of a part of the population, due to the intraspecies competition encounterings for food supply and deprivation and secondly-and most important- to an impairment of the ―ideal‖ ―intrinsic growth factor‖ g(N). This means that the ―actual growth factor-r‖ is different (smaller or bigger) from the g(N) (see below). This also assumes that the system has not been evolutionary adapted to conditions of food inadequacy. Besides this, the logistic model, remains basically an exponential model, since the main growth of the population takes place during its exponential "phase" (this model, is "growth saturated" as we`ll see below). The logistic model (logistic differential equation) was introduced in 1838 by Pierre Francois Verhulst, a Belgian mathematician and doctor in number theory. The differential equation of the cell growth of this model is as follows:

(for the solutions of the differential equation, see below). It is obvious that in this model the "intrinsic growth rate" is:

LOGISTIC GROWTH MODELS 2) If we now plot the g(N) as a function of N, we take the following figure.

Fig.9. â&#x20AC;&#x2022;Intrinsic factorâ&#x20AC;&#x2013; curve of Logistic Growth. This means that g(N) depends on the density of the members of the population, i.e. on the N(t) and this implies a decrease in the per capita growth-"yield" as the population increases. The g(N) versus N relation is a power law relation indicating self-similar growth.

3) The common form of logistic equation: also be written in this form:

can

, in which the term depicts a mortality rate proportional to the rate of paired intraspecific (intraspecies) encounters.

LOGISTIC GROWTH MODELS COMPUTATION OF Nmax AND dN/dt max OF THE CONTINUOUS LOGISTIC SYSTEM (MODEL) If we substitute in the above continuous logistic differential equation dN/dt = r(1-N/K)N , b=r/K, the we can write the equation in the form: dN/dt = rN -bN^2:

From the plotting of dN/dt in respect to N, we see that the Nmax is attained when the d [dN/dt] / dN = 0. Using a little algebra we compute that Nmax = r^2/2b and dN/dt = r^2/4b (Nmax is not the N=K -asymptotic value. In fact Nmax is a temporary value, because the system converges to N=K). The dN/dt max = r^2/4b, in fact indicates the minimum rate of extinction of the population (or tumour cells), at which the population (or tumour cells), will remain viable.

LOGISTIC GROWTH MODELS

Fig.10. The discrete (difference) Logistic equation The above discrete model, is an absolute analogous to continuous one. The fixed point of the "first iteration corresponds to the "convergence" of the continuous function to the K=carrying capacity of the continuous model. On the other hand, from the above fig. is evident that Nmax=1 corresponds to the Nmax = r^2/2b, of the continuous system. Yet again the Nmax is not the "solution" of the system, because the system in this value (point) is unstable and diverges away from it, converging instead to the fixed point = k ( ). Indeed the f``(x) at the Nmax value is f``(x) > 1 and it is a repeller. On contrary, the he f``(x) at the fixed point is f``(x) <1 and it is an attractor towards which the system converges. And this means that the system at the fixed point is stable and the fixed point is a â&#x20AC;&#x2022;solutionâ&#x20AC;&#x2013; of the system. This is the well known discrete logistic difference equation, which leads to the chaotic behaviour of tumour growth through the doubling period cascade. The tumour growth pattern depends in this context on the value of the parameters -r. Thus for 1<r<2 the population growth runs a doubling period of two, i.e. it oscillates between two points, for 2<r<4 the system exhibits a 4-period solution, oscillating between 4 points. Lastly, for values the growth attains a chaotic, that is it exhibits for small windows of r (narrow white perpendicular areas) the system exhibit regular periodic oscillations (solutions), after which they become aperiodic. The sequence of aperiodicity-periodicity-aperiodicity is 53

DIMITRIOS ANTONIOU B) THE DISCRETE MODEL (LOGISTIC MAP) OF TUMOUR LOGISTIC MODEL The above continuous logistic model is considered to be a continuous function of time. In reality this is not usually the case in population dynamics, where a discrete model, in the form of a "difference equation" will be more realistic. This because the tumour growth should be viewed as a discrete biological phenomenon, since tumour growth occurs in distinct "cell cycles" with characteristic "cell division times". This also means that the discrete model does take into account the "population cell death rate" and it is thus more realistic than the continuous one. This model is in fact is the mapping of the function of the population evolution of itself. In other words we follow the values of f(x)= x, in the usual manner, as depicted in the following picture, in absolute analogy to the above described discrete population model. So the discrete difference equation: f(x+1) = r x (1-x) is depicted as follows:

Fig.14. The Chaotic map of the Logistic model is actually a Probability Density Distribution function of its periodic points (see p. 202-213) We again want to emphasize that, we do not yet established a chaotic growth pattern of the tumours (logistic models, or Gompertzian), since we haven`t yet defined the appropriate parameter r > 4, but , according to the above, we still have indications of chaotic tumour growth, an indications which strengthens, by the data we`ll explore next, on this paper. We therefore can expect the tumour`s qualitative properties to be determined by the factors -r (intrinsic reproductive rate) and the carrying capacity K (or b=a/K). Parameters -r can be affected positively by growth factors as TGFs, PGFs, angiogenic growth

LOGISTIC GROWTH MODELS factors and by various cytokines (mainly IL-1 and IL-2-11). Inhibitory effects on -r can be asserted by defects in oncogenes, such as the p53 gene deficient or mutatant. Analogous effects we can expect and upon the rest tumour parameters of the logistic model.. THE AUTOREGRESSIVE TUMOURS The realistic tumour growth according to the discrete logistic equation, gives the first indications of the possibility of chaotic growth pattern of the tumours, since the analysis of the iterates of the map of the above discrete logistic equation, is well known that gives rise to various timebehaviours, such us: spontaneous extinction and regression, oscillatory growth into various limit cycles and finally to a period -doubling cascade to chaos. Do we have any, at least indirect evidence, that tumours can attain such a "chaotic" growth pattern as the next known figure?:

Fig.15a. Bifurcations and Doubling periods of the discrete Logistic model We actually know that many tumours show auto-regression, in other words the tumour cell population returns to its original (theoretically zero) number, as the above discrete model predicts, in its chaotic

GOMPERTZ GROWTH MODELS that system, namely the N, by a "power law" function. This implies that the growth of the tumour is self-similar (invariant) under population N scaling process. Moreover, it is also time-scaling invariant, because Gompertzian system is a time-series fractal growth system (see below). B.2) Gompertzian Curves

Fig.17. Gompertzian (dotted line) vs Logistic (complete line) functions and curves. Note that the Gompertz curve is asymmetrical with an inflection point at: < K/2.

FRACTAL GROWTH MODELS FRACTAL GEOMETRY AND POWER LAWS Self-similarity in Euclidean and fractal objects- Power Laws I) One dimension fractal: Fractal structure or growth means freescale self-similarity, namely structure (or property) invariance under continuous scale variance, or under continuous and/or arbitrary coordinate variance. For a smooth curve, an approximate length L(r) is given by the product L(r) = N r, where N is the number of straight-line segments of length r =1, needed to step along the curve from one end to the other. As the step size r down scaled to zero at increments of e=1/2, the L(r) continuously increase, in the sense that more downscaled straight-line segments rn are needed to cover the original smooth curve. Taking the logarithms of the number N(rn) of the scaled straight-line segments needed to cover the original smooth curve and of the rn = 1/(2^n) r (where n= the scaling step number), we find a relation as such:

(1) where Îł is constantly changing in relation to magnitude of the step size rn= 1/ (2^n) r. This continuous until a crucial size rn-c = 1/ (2^n) r is reached, below which the number of N(rn) increases proportionally to the length of each smaller rn= 1/ (2^n) r. In this situation the above equation becomes:

(2)

FRACTAL GROWTH MODELS

We also see that the Nn is a power function of the scaled original linear length, as follows: where e = scaling factor. The power law curve of Sierpinski`s triangle has the form:

Fig.4. Power Law curve of Sierpinski`s triangle: DH (=d) = 1.58â&#x20AC;Ś.

DIMITRIOS ANTONIOU SELF-SIMILARITY OF TIME-FRACTALS Tumours can grow in a time-series mode (like the non-autonomous Gompertzian) and in time-scale self similar mode, as follows: We have seen that the fractal self-similarity is expressed mathematically by the power law equation: , where N are the number of the self-similar objects, e the scaling factor of the scaled ―unit of measurement‖ of N and DH the Hausdorff exponent (dimension) of the fractal. Accordingly, if we scale the variant x of a function y=f(x) by a ―scaling factor‖ w and we find that it holds: y=y(wx)=(w^a)y (power law), we say that the y=f(x) shows self-similarity under scaling. The exponent a depends on the ―dimensions‖ of the fractal y ―object‖ and if a is an irrational non-integer, then the y is self-similar and fractal ―object‖, as the following equation shows:

In time-fractals, the above function y, is substituted by the statistical properties S of a function f(t), the scaled variable x by the ―time-t‖ and the scaling factor w by the scaling factor ―time-window‖ Δt = τ of the original ―time-unit‖. Statistical properties can be ―mean value‖, ―autocorrelation function‖, ―standard deviation‖,etc (p.237-243). Time-window in general in physics is a function whose all values outside its boundaries are zero. We thus use the following method to establish the self-similar and/or fractal nature of a times-series: We first arbitrary chose a «time-window-interval-τi», regarded as ―unittime‖ onto which the statistical property S1 is referred to (S1 is completed over time-duration τi). Then we arbitrary chose a timewindow-τii, been a fraction of τii and thus been used as the scaling factor of τi. The τii is again regarded as ―time-unit‖ onto which the statistical property S2 is referred to (S2 is completed over timeduration τii). Finally, we find the relation between the ration of SI / SII and τi / τii. The relation would be of the power law nature if the timeseries is self-similar, which further means that it remains «invariant under time-scaling», as it could be f.e. an in time-evolving encephalogram, or a fractal music frequency distribution interval (see next page):

FRACTAL GROWTH MODELS

Fig.5. Self-similar time-fractal. The magnified time-window n1 acquires equal statistical properties with window n2.

FRACTAL GROWTH MODELS FRACTAL DENSITY Suppose that F is a subset of

. By analogy of what we said in p.69-

71, we define as the measure of F the limit value: , which is called: the s-dimensional Hausdorff measure of F (of either length or surface area). Obviously for Euclidean sets is an integer number <0< , with s = integer (1,or 2, or 3), while for fractal sets the can (and usually is) 0 or and s= non-integer and irrational number. Now let consider s-sets F, i.e. Borel sets of Hausdorff dimension s, with positive finite and let F a Borel subset of the plane . The density of F at x is:

(1) where Br(x) is the closed disc of radius r and centre x and the ― ‖ symbol stands for ―intersection‖ of two sets (Set theory). The classical Lebesgue density theorem tells us that, for a Borel set F, this limit exists and equals=1, when x belongs to F and 0 when x does not belong to F, except for a set x of area=0. Similarly, if F is a smooth (non-fractal) curve in the plane and x is a point of F (other than an endpoint), then is close to a diametrical chord of F for small r and

(2) If x does not belong to F this limit is clearly 0. Density theorems as these tell us bow much of the set F, in the sense of area or length, is concentrated near x. Now we ask:: If F has a fractal dimension s, how does the s-dimensional Hausdorff measure behave as r--->0 ?. Or equally: Does an analogue of the Lebesgue theorem holds for fractal sets (objects)?. We look at this question when F is an s-set in R^2, with integer or non-integer s of: 0<s<2 (s=0 sets are just finite sets of points, whereas is essentially area, so if s=2 we are in the Lebesgue density situation of equation (1)).

FRACTAL GROWTH MODELS The above dynamics satisfy the dynamic scaling ansanz of Family and Vicsek and hold for many rough fractal physical systems, although in some cases they do not hold entirely in that way. Thus, in what is called super-rough systems, the "global exponent -a" attains a value a> 1. In these cases the local surface width w(l,t) does not saturate as depicted above, namely, as:

but instead it crosses over to a new behaviour in the intermediate regime exponent

, characterized by a different growth , and

, where

A scaling showing these features is called "anomalous scaling". Before we move on this article, we need to define the term: "structure factor": Structure factor, is the Fourier Transform of the spatial distribution of the global width:

. The interface Fourier transformation gives us the power spectrum of the Structure factor, according to dynamic ansanz scaling of Family and Vicsek:

where s is the structure factor, which shows the scaling behaviour:

DIMITRIOS ANTONIOU EVOLUTIONARY ADVANTAGES OF THE (MULTI) FRACTAL TUMOUR AND ANIMAL GROWTH BY OPTIMAL ADAPTATION "The beauty and necessity of breaking the linear proportionality for a non-linear one". The above described fractal architecture attained by the tumour, provides the latter with certain evolutionary advantages, which we make clear with an example. A simple case is that of two similar weights hung up by two similar wires. The forces exerted by the two weights on the wires are proportional to their masses and the latter to their volumes and therefore finally to the cubes of their several linear dimensions I^3, including the diameters of the wires. The weight capacity (weight sustainability) of the wires is dependent on the exerted weight / per unit area of their cross-section surface area. But the cross-section areas of the wires varies as the square of their said linear dimensions I^2. Therefore the weight strain, exerted and sustained per unit area of the cross-section area of the wire increases as the ratio of the cube power of the linear dimension of the weights (their volumes) and the square power of the linear dimension of the wires (their cross-section surfaces). This means, that the larger the structure, the more severe the strain on the wires becomes and that the wires are less capable of supporting it:

where l is the linear dimensions. This equation shows that a statistical property, f.e. the exerted weight / per unit area, is varied in a selfsimilar power-law fashion. This implies the followings: It often happens that of the forces in action in a system some vary as one power and some as another, of the masses, distances, or other magnitudes involved. The "dimensions" remain the same in our equations of equilibrium, but the relative values alter with the scale. This is known as the "Principle of Similitude", or dynamical similarity and it and its consequences are of great importance for the evolution of livings. To come back to our example, lets consider now this: The strength of an iron girder obviously varies with the cross-section of its members and each crosssection varies as the square of a linear dimension; but the weight of the whole structure varies as the cube of its linear dimensions. It follows at once that, if we build two bridges geometrically similar, the larger is the weaker of the two and is so in the ratio of their linear dimensions. 98

DIFFUSION GROWTH MODELS PRELIMINARIES FOR THE STUDY OF DIFFUSION TUMOUR GROWTH MODELS Î&#x2018;) DIFFUSION-LIMITED AGGREGATION (DLA) a) "Discrete - DLA -model The diffusion-limited aggregation model (DLA) provides a convincing simulation of the tumour growth. The model is based on a lattice of small squares. An initial square is shaded to represent the "cathode" and a large circle is drawn centered on this shaded square. A cell is released from a random point near the perimeter of the circle, and it, by performing a Brownian motion, eventually reaches at a square neighbouring the one previously shaded. The new square is then shaded as well. As this process is repeated, a connected set of squares grows outward from the initial one. Running the model, say, for 1000 shaded squares, gives a highly branched picture that resembles a branching tree (see fig. below).

Fig.1. Diffusion-limited aggregation model (DLA) We then perform "box-counting" to estimate the dimension of this cancer cell assembly.

103

DIFFUSION GROWTH MODELS FRACTAL DENSITY Suppose that F is a subset of

. By analogy of what we said in p.69-

71, we define as the measure of F the limit value: : which is called: the s-dimensional Hausdorff measure of F (of either length or surface area). Obviously for Euclidean sets is an integer number <0< , with s = integer (1,or 2, or 3), while for fractal sets the can (and usually is) 0 or and s= non-integer and irrational number. Now let consider s-sets F, i.e. Borel sets of Hausdorff dimension s, with positive finite and let F a Borel subset of the plane . The density of F at x is:

105

DIMITRIOS ANTONIOU THE DIMENSIONLESS MODEL If we take the non-dimensional variables:

(11.9), then the (11.6) becomes:

(11.10),

where:

(11.11). Theoretically we consider the growing tumour cells to be originated from the center of a sphere, which represents the "core of the tumour". If the center is in xo position and the maximum concentration of cells in xo is a, then the concentration of cells around xo at, is distributed according to the Gaussian normal distribution equation:

(11.12). In homogeneous tissue (Dg=Dw), if p=constant and with initial delta function source of No cells at x=0, the (11.3) has the solution:

(11.13), where r is the axially symmetrical radial coordinate,

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DIFFUSION GROWTH MODELS

A short note on Fisher-Kolmogorov wavespeed equation: (11.38) Fisher proposed (1936) the folowing second order non-linear parabolic partial differential diffusion equation, as a model of the propagation of a mutant gene in the human population with a selection intensity s and in many other biological phenomena (see bibliography).

(1) There are no solutions for wave velocity c < 2, whereas for c>=2 the solutions are of the general form: u(x,t) = u(+-x+ct) (see below eq.2). Kolmogorov showed (1937) that, if the initial datum satisfies 0<=u(x,0)<=1, u(x,0)=1 for x<a and u(x,0)=0 for x>b (that is the wave switches between the two equilibrium states 0 and 1, i.e. in bounded conditions), the solution approaches a travelling wave of speed=2. For the special case of:

the solutions of (1) are given:

(2)

where C is an arbitrary constant >0. The above solutions are of Painleve`s transcendents type, that is, the wave shape for a given wave speed is not necessarily unique (i.e. the solutions admit only poles as movable singularities). There could also be solutions with c>2 under appropriate conditions. Andrey Nikolaevich Kolmogorov (1903 â&#x20AC;&#x201C; 1987) was a Russian mathematician and Sir Ronald Aylmer Fisher, FRS (1890-1962) was an English statistician, biologist and geneticist (see the original Fisher`s paper in:

http://digital.library.adelaide.edu.au/dspace/handle/2440/15125 Paul Painleve was a French mathematician and Prime Minister (18631933).

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TUMOUR INVASION IN HUMAN BRAIN Now a few data concerning the simulation computer program used in this study: The program is the EMMA (Extensive MATLAB Medical Analysis) developed by Collins (1998) at the McConnel Brain Imaging Center (Montreal Neurological Institute) and it is a program for manipulating medical images. The Brain Web database, which provides this program http://www.bic.mni.mcgill.ca/brainweb/ , was created using MRI simulator and defines the locations and distribution of grey and white matter in the brain in 3-d spatial dimensions on a 181 x 217 x 181 grid (see next fig.1). It was created to visualize frozen images in MATLAB so that they can be manipulated and studied.

Fig.1. Distribution of white and grey matter in brain. For our theoretical two dimensional (computer) simulation of tumour invasion in brain, we allow a 10-fold variation in p (growth rate) and D (diffusion coefficient) to simulate different tumour grades. The numerical simulation allows us to track the invasion of virtual tumours of any initial size and distribution from any site. We give some examples of the simulation figures further down of this paper, interpreting also their meaning. Our simulation generally indicates that the invasion is greater at brain areas dominated by white matter (corona radiate and corpus callosum). Especially the corpus callosum greatly facilitates the spread of the tumour to the contralateral hemisphere. The model also showed that the tumour`s growth is affected to a great extent by the "anatomical barriers" to its spread, as are the ventricles. The ventricles in fact, contribute, from one point of

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DIMITRIOS ANTONIOU

THEORETICAL SURVIVAL TIME IN HOMOGENEOUS BRAIN TISSUE TUMOUR (untreated cases of virtual tumours) In case of constant p and D(x)=D, the solution of equation

, is:

(11.3),

(11.13), where No the initial delta function source of No cells at xo and r is the axially symmetric radial coordinate. Clinically the "lowest critical concentration" c* of tumour cells, sufficient to produce a detectable resolution contrast with the normal brain tissue and therefore detectable by the CT scans, is c* = 40.000 cells/ cm^2. If the critical for its detection low threshold density of the tumour is c* (i.e. if c(x,t)=c*), then the corresponding to c* axially radial coordinate is r*, as computed from (11.13):

(11.14). The last expression is just the asymptotic form of the radial travelling wave of the axisymmetric Fisher-Kolmogorov equation and spreads with diffusion velocity: . So, if the tumour is identified when it has the radius ―r-detect‖ and the tumour is fatal when the radius ―r-lethal‖ is reached, we can approximate the untreated survival time by:

. This shows that D and p are both important parameters in determining survival time: increasing either D or p will decrease the survival time. The average radius at which a tumour is identified is 1.5cm and the lethal is 3.0cm (generally). Within these radii range the approximate

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DIMITRIOS ANTONIOU

Fig.6. Virtual tumour introduced in the temporal lobe of brain, denoted by the asterixes, for three cross-sections through the 3-d tumour, namely sagittal, coronal and horizontal which intersect the tumour site. Pictures on the left columns show tumours at diagnosis, while the right columns show tumours at patient`s death. The dark line defines the edge of the tumour detected by a CT scan in which the "standard low density threshold of detection" is c*= 40.000 cells /cm^2. The wavelike lines define the tumour boundaries calculated out to â&#x20AC;&#x2022;sensitive thresholdâ&#x20AC;&#x2013; of 5% of the "standard low density threshold of detection".

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DIFFUSION GROWTH MODELS

Fig.7. Simulation of virtual tumour invasion of a high grade glioma in the superior fronto-parietal cerebral hemisphere : (a), (b) at diagnosis; (c), (d) at death; (a), (c) the dark solid lines define the tumour spread as seen by standard modern CT scans and calculated by our "standard low density threshold of detection" of c*= 40.000 cells /cm^2; (b), (d) the wave-like lines define the tumour boundaries calculated out to â&#x20AC;&#x2022;sensitive thresholdâ&#x20AC;&#x2013; of 5% of the "standard low density threshold of detection".

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DIFFUSION GROWTH MODELS

(11.49), for tr < t < (r/Rr) tr. The following fig.12 gives the time scale of detection of the "least detectable" tumour radius r* with and without resection.

Fig.12. The graph gives the time scale of detection of the "least detectable" tumour radius r* with and without resection, in homogeneous tissue, for virtual brain tumour models. TUMOUR VIRTUAL MODEL IN HETEROGENEOUS TISSUE AFTER SURGICAL RESECTION If we take the non-dimensional variables:

as in (11.9), then the (11.10) becomes:

(11.50), (where p=1) and

(11.51).

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DIFFUSION GROWTH MODELS

Fig.14. Simulation of virtual tumour invasion of high grade glioma in the inferior fronto-parietal lobe following "gross surgical removal" (left column) and "extensive resection" beyond the CT detectable margin of the tumour (right column)

141

DIFFUSION GROWTH MODELS CHEMOTHERAPY IN HOMOGENOUS TISSUE AND IN ONE CELL POPULATION MODEL In this model-along with the effectiveness of chemotherapy-it is tested the reliability of the diffusion model for accurately expressing the growth features of in vivo grown gliomas, using the principles of method ii of p.117. The present model applies to a patient with an anaplastic astrocytoma treated three years earlier with X-ray irradiation. The tumour recurred at the time of initiation of the present study. The patient survived for 12 months after the recurrence was diagnosed. During his final year the patient received two different chemotherapies. The first one was a course of six drugs, given over 15 days and repeated every six weeks to allow for recovery of bone marrow and applied in total five times. The second course consisted only of two courses of cis-platinum given at monthly intervals. In addition, the patient received neutron irradiation during the last three months. The tumour cell density c(x,t) (expressed in terms of number of cell-nuclei / mm^2) was determined by image analysis of a biopsy obtained at the time of the diagnosis of the recurrence and was averaged over 5 CT scan section images. During the treatment, the tumour`s cell density evolution was being assessed in terms of the tumour`s detected area evolution over time, with serial CT scans. The pre-treatment cell density (c(x,t)) was used as a starting (initial) condition for the model. A second evaluation of tumour cell density was performed at autopsy material, with the same as above method. The autopsy tumour cell density was used as the threshold in the model, over which the CT area was determined to be tumorous. The threshold was taken to be 40% of the autopsy tumour density. Our model assumes that the tumour consists of one cell type. We also assume homogenous brain tissue conditions i.e. the tumour growth and the diffusion coefficient are constant. The basic mathematical model (before the initiation of the chemotherapy) is thus governed by the standard diffusion equation:

(11.3). In order to quantify the effect of chemotherapy, we assume that the cell rate death due to chemotherapy can be modelled by a linear removal rate K(t) c(x,t), where c(x,t) is the density (concentration) of

143

DIFFUSION GROWTH MODELS CHEMOTHERAPY IN HOMOGENOUS TISSUE AND IN TWO CELL POPULATION MODEL But our hypothesis of a single cell type in the previous tumour model was not fitted with the clinical data, concerning the sharp increase of the tumour at the end of the first chemotherapy. Therefore, we had to alter the model, to include in it two different cell types with "initial" densities c1(x,t) and c2(x,t) respectively. The first cell type c1 is sensitive to both types of chemotherapy, whereas the second one c2 is sensitive only to second type of chemotherapy (cis-platinum). The first type c1 comprises 90% of the total tumour cell burden at the first scan prior to commencement of the chemotherapy. The second cell c2 type is considered as a distinct cell type on its own right and NOT as mutation per generation of the first type during the chemotherapy period (such a model will be studied in the next section). Our model assumes also that the two cell types have the same diffusion coefficient D, but different growth rates p1 and p2. Therefore our model now can be represented by the following system of differential equations, with the corresponding boundary and "initial conditionsâ&#x20AC;&#x2013;:

(11.56). Here we assume the first cell type c1, is sensitive to both types of chemotherapy, whereas the second one c2 is sensitive only to second type of chemotherapy (cis-platinum). Therefore we take:

(11.57).

145

DIMITRIOS ANTONIOU MODELLING TUMOUR POLYCLONALITY AND CELL MUTATION In this model we assume that the tumour consists of two cell types, which are not independent, but instead, they are connected by mutational events which transfer (mutates) cancer cells from the originally present cell population v to cell population u. We also assume that the two cell types have different diffusion coefficients D2 > D1 and different growth rates p1 > p2, i.e. we assume that the cell type u has a high diffusion coefficient D2 and low growth rate p2, whereas the cell type v has a low D1 and high p1. We also assume homogeneous brain tissue conditions. Under these conditions, the total cell density c (=c(v)+c(u)) becomes the vector of the two components of the cell densities of the two cell types, while the diffusion coefficients D1,D2 and the growth rates p1,p2 become now diagonal matricies of diffusion coefficient and growth rates respectively. The total number of cells at a given point in space and time c(x,t) is given by the sum of the components of the c vector. We also define as T the matrix which transfers the mutated cells from one cell type to the other. Therefore we take the following general diffusion equation:

(11. 60). The cell density c and the matricies of the various scenarios of the combinations of p and D, are as follows:

From (11.60), for each cell type we take the following equations:

(11.61),

152

DIFFUSION GROWTH MODELS taking as boundary and initial conditions the ones depicted in equations (11.8) and (11.12) respectively. As we`ve said, we define with v the concentration (cell density) of the more rapidly proliferating cell type and with u the concentration (cell density) of the more rapidly diffusing cell type, i.e. D2 > D1 and p1 > p2. We further assume that at first only the v cells exists and that, by a small mutation probability parameter k << p1, v cells mutate and become u cells. We further introduce the following non-dimensional variables:

(11.62),

(11.63).

where vo = (integral) â&#x2C6;Ťf(x)dx, the total (and solitary) original cancer cell population. Growth is measured on the timescale of the v proliferation and diffusion is on the spatial scale of the v cell diffusion. Then equation (11.61) becomes:

(11.64),

(11.65).

153

DIMITRIOS ANTONIOU Convolution Theorem: Definition:

We now derive from the afore mentioned time convolution theorem. Suppose that g(x) = f(x) h(x). Then, given that Ω{g(x)} = G(s), Ω{f(x)} = F(s), and Ω{h(x)}= H(s), we take: G(s) = Ω{f(x)

h(x)}

= Ω { ∫ f(b) h(x -b ) db} = ∫ f [ ∫ f(b) h(x -b ) db] exp(-i 2pisx) dx = ∫ f(b) [ ∫ h(x -b) exp(-i 2pisx) dx] db = H(s) ∫ f(b) exp(-i 2pisb) db = F(s) H(s). This extremely powerful result demonstrates that the Fourier transform of a convolution is simply given by the product of the individual transforms, that is: Ω{f(x) h(x)} = F(s) H(s). Using a similar derivation, it can be shown that the Fourier transform of a product is given by the convolution of the individual transforms, that is: Ω{f(x) h(x)} = F(s) H(s) This is the statement of the frequency convolution theorem.

156

PREDATOR-PREY SYSTEMS I.A. PREDATOR-PREY SYSTEMS (Lotka-Volterra equations) The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems, in which two species interact, one as the predator and the other as its prey. They were proposed simultaneously but independently by the American mathematician, physical chemist, biophysicist and statistician Alfred J. Lotka (1880-1949) in 1925 and the Italian mathematician and physicist Vito Voletrra (1860-1940) in 1926. The "classical" Lokta-Volterra predator-prey system of animals, microbes or cancer cells growth is basically an exponential growth model, with the addition of encountering conditions between the species populating the system. In particular, the system assumes that there is unlimited food supply to prey animals, which therefore could (potentially, i.e. in the absence of predators) grow exponentially (there is no intra-species competition for food amongst preys). This guarantees a theoretical unlimited food supply to predators too, which could thus also grow (potentially) exponentially. But because predators consume preys, the food supply of the former is not really unlimited but rather self-limited and directly dependent on their own number. Thus, the whole system can be described by assuming that the ―growth rate of preys‖ is proportional to their population (exponential growth) and the ―death rate of predators‖ is proportional to their own population too (exponential decay). The whole system can also be described as assuming ―no intra-species competition‖, but with ―interspecies encounters‖ (see also p. 273, 275 & 277). We examine here the "classical" predator-prey system, since the Logistic predator-prey system can easily be derived by combining the former with the "classical" logistic model (see below). We call "prey" an animal species, as for example the rabbits (R) whose unlimited food supply is ample food. The growth rate of the prey species is exponential, i.e. if R(t) is the number of rabbits in time, the growth rate of them is expressed by the non-linear differential equation: Growth Rate of Prey:

(1.1), where k is a positive constant.

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DIMITRIOS ANTONIOU

Fig.1. Phase portrait of a Predator-Prey system (Lotka-Volterra equations) d) If at time t the number of wolves is W=40 in point Po (well below their "equilibrium" value Weq= 80) then, from the above equation dR/dt we get dR/dt=40. Therefore, we have to move towards the anticlockwise in the phase portrait (see the figure 2 below) to follow the evolution of the system. Following the system this way we see that, as the number of rabbits increases from point Po to the maximum number P1= 3000 at point P1, the number of wolves increase too, from point Po to point P3 (clockwise now). But as the product RW increases, as above, the number of encounters increase too and therefore the number of rabbits reaches a maximum at point P1=3000, because now there is sufficient large number of wolves to eat them. The number of rabbits starts to diminishes from point P1 to point P2 (where it reaches again the initial number of P2=Po= 1000). The wolves, at the same time, have already increased to their maximum number P2= 140. From the point P2 now, both species start to decrease to point P3. From the point P3 the number of rabbits starts again to increase towards the point Po. At the same time, the number of wolves continues to decrease from P3 to Po too. At point Po the system has again reaches its starting point and the cycle starts again.

168

DIMITRIOS ANTONIOU I.B. PREDATOR-PREY MODELS OF TUMOUR-IMMUNE INTERACTIONS These mathematical models are based - in broad terms - on the above predator-prey system, where the place of rabbits is taken by the "malignant cells" and the place of wolves is taken by the "lymphocytes-immune system cells" (macrophages, Natural Killer cells, T-cytotoxic cells, IL-2 activated T lymphocytes etc). In biological reality anyway the above "roles" of predator and prey of lymphocytes-malignant cells could be inversed. If we denote X(t) and Y(t) the number of malignant and lymphocyte cells respectively, we take the following non-linear and oscillating system of coupled non-linear differential equations, in analogy to the above described predator-prey system: The growth of the malignant cells (prey species) is exponential, i.e. if X(t) is the number of cancer cells in time, the growth rate of them is expressed by the non-linear differential equation: Growth Rate of cancer cells:

(1.1), where k is a positive constant. We next call Y(t) the lymphocytes (predator species). If the number lymphocytes-predators as a function of time is Y(t), then, the rate of growth diminishes exponentially in time, i.e. the rate of deaths is proportional to their own number. The above are expressed in the following non-linear differential equation: Death Rate of Lymphocytes:

,where r is a positive constant

(1.2).

Finally we assume that the number of encounters between lymphocytes vs cancer cells (predator vs prey) are proportional to both populations and are therefore proportional to the product XY. The

172

DIMITRIOS ANTONIOU II.A. LOTKA – VOLTERRA COMBINED WITH LOGISTIC GROWTH SYSTEMS (INTERSPECIES ENCOUNTERS AND INTRASPECIES COMPETITION MODELS) Despite the success of the above described ―classical‖ Lotka-Volterra systems to predict the evolution of many coupled biological systems, more sophisticated models have been evolved as complementary to the "classical" predator-prey system. Such a system is the Lotka-VolterraLogistic system, which assumes a ―classical‖ Logistic growth of the prey population, complemented with encounters between the prey and predators. Thus, one way to describe more realistic systems than the "classical" inter-species competition Lotka-Volterra system (which assumes a potentially - in the absence of predators- exponential growth of preys and exponential death of predators and finally, no intraspecies competition for food supply amongst the preys) is to assume that in the absence of predators, the prey population grows according to Logistic system, i.e with intra-species competitions for food nutrients with carrying population capacity K, in conditions of limited blood supply of nutrients. Then the above depicted Lotka-Volterra equations are replaced by the following system of differential equations:

(see also p. 275, 277). MORE SYSTEMS OF TWO SPECIES (OR TWO CELLS) COMPETITIONS Systems have also been proposed to describe and predict population levels of two species that compete for the same resource, or cooperate for mutual benefit. Such models are expressed with the following systems of differential equations: a) dx/dt = ax - bx^2 + cxy, dy/dt = zx + g xy, where: ax-bx^2=y and a,b,c,z,g, are positive constants: Cooperating system: b) dx/dt = ax - bx^2 - cxy, dy/dt = zy - gy^2 - wxy, where: a,b,c,z,g,w are positive constants: Competing system.

174

PREDATOR-PREY MODELS II.B. LOTKA-VOLTERRA COMBINED WITH LOGISTIC MODELS OF TUMOUR-TUMOUR INTERACTIONS An important consideration in the modelling of polyclonal cell tumours is the inter- and intra-species competition, or (indirect) encounters for common nutrients and growth factors between the different types of tumour cells of the same tumour. The simplest model satisfying the above conditions, is a Lotka-Volterra model, which assumes inter-species (indirect) encounters and competitions amongst the different types of tumour cells for blood nutrients, i.e. a predatorprey competition system combined with a Logistic Growth model of tumour cells of the same type. This assumes an intra-species competition for blood nutrients with carrying capacity K, in conditions of limited blood supply of nutrients (see also p.275, 277). If we assume that the variables n(t) and m(t) represent the population numbers of the different tumour cell types n and m at time t, we can write the model as follows:

where rn and rm represent the growth rates and Kn and Km the carrying capacities of the cell populations n and m respectively. The inter-species competition for resources are reflected by the gn and gm. The parameters are all positive. It is well known that apart from the trivial steady state, So = (0,0), the Lotka-Volterra system has also these three steady states: S1 = (Kn, 0), S2 = (0, Km ) and the â&#x20AC;&#x2022;coexistentâ&#x20AC;&#x2013; (combined) state:

Furthermore the solutions are bounded and there are no limit cycles. 175

ORDINARY DIFFERENTIAL EQUATIONS Ordinary differential equations An ordinary differential equation (ODE) is any statement linking the values of a function f(x) to its derivative and to a single independent variable, for example:

The order of the equation is n, the degree of the highest derivative that appears in the equation. The solution of an ODE is the function y=f(x) that satisfies the equation for every value of the independent variable x. Linear equations have the special form:

In linear equations no multiples of other nonlinearities in y or its derivatives occur. F.e. the equation: dy/dt = t^2 – y, is a first order linear and time-dependent equation, because there is no non-linearity on its solution function y = f (t) (on the dependent variable y) and it explicitly depends on t. On contrary the equation: dt/dt = y^2 – t is a first order non-linear and time-dependent equation, because it contains the non-linear term y^2 and explicitly depends on t. The coefficients ci (i=1,2,3…) may be functions of the independent variable t. The case of constant coefficients is of particular importance to stability analysis and can be solved completely.

179

DIMITRIOS ANTONIOU Time-independent (time-invariant) function (system) Now let`s consider for clarity our know autonomous logistic system, whose autonomous differential equation is this: dy/dt = y (1 – y /K) = y – (y^2)/K. A system described by such an equation should be unfolding at some inherent rate independent of the clock time or the time at which the process began, because the equation does not depend explicitly on the dependent variable t. The direction field of the above equation is the following:

Fig.3. Structure of the direction field of the logistic equation: dy/dt = y (1 – y /K). For each pair of value (y,t), line segment whose slope is y (1 – y /K) is drawn. Number 1 represents the carrying capacity.

Fir.4. Solution curves are constructed by maintaining tangency to the directions shown in fig.3. Along each of these line loci of constant values dy/dt=Ki, the tangent solution lines are parallel and of constant slope =Ki. Number 1 represents the carrying capacity.

182

LINERIZATION OF NON-LINEAR SYSTEMS The significance of the steady states of non-linear systems Consider the autonomous (time-independent) functions f(x) and f(t). Their steady states

feature the property that df/dt at

point is zero: state is stable if:

. A steady

(this is a type of change in variable). A steady state is termed stable if spatial neighboring states are attracted (converge) to it and unstable if the converse is true (see p.199). Proposition: We know (is not proven here) that the non-linear systems close to their steady states behave nearly like their linear analogues (counterparts). Therefore we can investigate and determine the features of the steady states of the linear systems and from them derive conclusions (predictions) for the nonlinear ones close to their steady states too, by assigning to each of the latter a respective linear steady state model. Properties of a system of two LINEAR ODEs

187

LINERIZATION OF NON-LINEAR SYSTEMS

Thus close to

(0,0) the non-linear system (1) behaves much

like the linearized version:

(4a).

This is a saddle point (see fig.3). Similarly, close to non-linear system behaves much like the linearized version:

(1,1) the

(4b). This is an unstable node (see fig.3). Using the above reasoning, we obtain the three upper figure-eigenvector possibilities of fig.3 of our system of two linear ODEs.

Fig.3. Local behavior (phase portrait) of a linear system and a (linearized) non-linear system close to its steady state for real eigenvalues 位. Upper row: Eigenvectors (p.336). Lower row: Solution curves. For real eigenvalues 位1,2 the eigenvectors vi (i=1,2) are directions on which solution curves travel along straight lines towards or away from the steady state (0,0). If 位1,2 are both positive, the direction of flow along vi is away from (0,0), whereas if 位1,2 are

191

DIMITRIOS ANTONIOU Τhen we can equivalently construct an β γ parameter phase plane which shows the possible behavior of both linear and non-linear systems near their steady states, as shown in the following fig.4.

Fig.4. General flow behavior (solution curves) of linear and local behavior of non-linear systems close to the steady states of the latter . The β γ parameter plane showing the 6 regions in which the steady state is an unstable (a) or stable (c) node, an unstable (d) or stable (f) spiral, or a saddle point (b,b), according to the values of β and γ. Finally the set (e) of disjoint neutrally stable concentric closed curves (trajectories) encircling the neutral stable center (origin) is structurally unstable, in the sense that small changes may disrupt its balance (steadiness) (p.197). The a-e solution curves could also be level (contour) flow curves of a surface flow of a conservative system z = f(x,y) (see p.267 – 282, 343-8). We here emphasize the obvious: In the steady state the flow is stagnant and its existence necessitates that the system is time-independent (autonomous). We thus see that there are only a few (six) possible stable and unstable attractors (states) that a non-linear system can attain in the phase plane close to its steady states, in analogy with the possible steady states a linear system can attain. Returning back to the Jacobian (3) and (3a) we see that for (0,0), β=0 and γ = -1. So from fig.4 we see that the (0,0) is a saddle point.

194

DIMITRIOS ANTONIOU

tď&#x192; , rendering the limit cycle either stable or unstable respectively and secondly its structural stability. A dynamical iterating system consisting of two non-linear DIFFERENCE Equations (see p.200).

The stability conditions of a system of two non-linear DIFFERENCE equations are tabulated through its linear counterpart, as shown in details above, i.e. by linearizing the system of the non-linear equations according to the following linear system of two linear difference equations: Linear System of two iterating Difference Equations

198

DIMITRIOS ANTONIOU II) Non-linear Equation LOGISTIC (non-linear) DIFFERENCE EQUATION The solution of the difference equation is the logistic solution curve of fig.1.

Fig.1. Logistic (non-linear) solution curve and derivative curve of a continuous logistic equation (dynamical system).

Fig. 2. The parabola f(x) is used to graph the successive values xn of the non-linear difference equation , which is indeed the â&#x20AC;&#x2022;first generation curveâ&#x20AC;&#x2013;. This is the phase portrait of f(x) and represents the discrete logistic equation (system), or logistic mapping.

202

MAPPING –LOGISTIC EQUATION Mapping, Iteration, difference equations, dynamic systems Mapping is any transformation of an interval to another through some transformation function, for example the transformation of the domain of the independent values (variable) x, to the domain of dependent values (variables) through the transforming function f: xf(x). The transformation (mapping) can be a continuous process, like f.e. the iteration of the derivative df/dx of one function f(x), or the Jacobian transformation of one function, or it can be discrete, such the iteration of a function f(x): xn+1 = f(xn). In p.32 we describe the iterationg process as this in which the dependent value of the previous step n becomes the independent value of the next step n+1, or: xn+1 = f(xn), like f.e.our known logistic function (difference equation) F(x) =x(1-x).

. If the iterating process reaches a point such that:

The point f(x)=x is a fixed point, which denotes the equilibrium state of the iterating system. This is further explained next: Let take the following graph of the non-linear quadratic (logistic) function: F(x)=rx(1-x), for 2<r<3.

Fig.1. Mapping (graphical analysis) of : V—>V mapping of the function Fr(x) =rx(1-x) on itself, with r =2.8.

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DIMITRIOS ANTONIOU

Fig.2. The graph of

of Fr(x) = rx(1-x), with

We`ve just seen that the bifurcation of

. was followed by two

results: i) The stable (attracting) fixed point c of period 1 of became unstable (repelling) periodic point in

and ii) the generation

of two new stable (attracting) fixed periodic points of period 2 in In other words, as the fixed point c of the

loses its ―attractiveness‖,

the period two orbit (of the periodic points b,d) of By further increasing the r,

acquires it.

loses the stability of its fixed points b,

d to other new stable (attracting) periodic points of while

of period 4 etc.,

maintains all the previous fixed points of its predecessors (all as unstable now). We thus follow the evolution of a

sequence of bifurcations of    etc. Each bifurcation occurs at different values of r, which (the r-s) constitute Feignebaum`s sequence of r-s (see p.52-54). Each time a transition point of bifurcation value is reached, some new qualitative behavior of the respective F is established (see next fig.3).

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MAPPING-LOGISTIC EQUATION

Fig.3. Bifurcations and Doubling periods of the discrete Logistic model The Feigenbaum constant δ =4.6620…depicted in the above figure 11, is actually showing that the parameters rs of the logistic difference equation are changing in a time-scale independent mode, characteristic of the self-similar processes. The process of generation:    etc. should be conceived as the result of the (iterating) composition of two functions, defined as: We denote the composition of two functions by F(x) o G(x) = F(G(x)) (see below). Therefore we have: where

is the ―bifurcating factor‖, that is the function F with a

specific value of factor r at which the bifurcation of to occurs. The bifurcating value of r is defined by the Feigenbaum sequence. Note that r is the intrinsic growth factor of the biological system in concern.

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MAPPING-LOGISTIC EQUATION

In words: The graph of that the graph of implies that

has exactly

humps in V and it follows

crosses the line y=x at least has at least

in V consists of

times. This

fixed points, or equivalently the periodic points of period n. The periods

of the periodic points in V are all of power of two (periods=

(For the significance of orbits of period 3 in the mapping see below Sarkovskii`s theorem). Studying

(x)

Fig.7. The graph of At first, we see that the phase portrait of the function Fr(x) does give information about its first iterate : xf(x), which is indeed the ―first generation curve‖, but gives very little information about subsequent bifurcated iterations, namely the iteration functions of higher order such as the as etc, or the ―second and third generation curves‖ etc. (composite functions F o G -see below)

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MAPPING-LOGISTIC EQUATION

Study of

then reveals that its second iteration function

have a periodic point between the interval

(x) -

should

(x) on x-axis.

Some useful Definitions Composition of two functions: We denote the composition of two functions by F(x) o G(x) = F(G(x)). The n-fold composition of F with itself (bifurcating iteration, or self-mapping bifurcation), recurs over an over again in the sequel. We denote this function by . The sign does not mean that the f is raised to nth power, nor does mean the nth derivative of f, which we denote by

. If

exits, we write:

Important Note: We must distinguish the sequence



etc. of the various bifurcated iterates of the function F, which is the result of the composition of two functions, from the sequential images (points) f(x),

(x),

(x) of every x which are emerged in the

iterating process inside the same iterate function , or , or etc. and which (sequential points) always end up in a fixed point F(x) = x, f.e. F(

(x)) = x inside each distinct function

, or

etc.

Fixed points: We return now again to fig.1 and we define: The point xo is a fixed point for F, if . Thus the fixed point is located at the crossing point of the line y=x with the F(x). Note that the fixed points of

(xo) =xo, are maintained as such and in the following

iterate functions

, or

etc. So we can say that the fixed points

never move under iteration and this because: Since follows that

(xo)=

(

(xo)) =

(xo) =xo, it

(xo) =xo and in general

. In that sense, fixed points are periodic points. For example, both 0 and 1 are fixed points for and . Similarly, 0 is a fixed point for S(x) =sinx. Accordingly the set of iterates of a fixed point form the periodic orbit of the fixed point.

213

CHAOTIC MAPPING CHAOTIC MAP Of special interest is the case of the quadratic (logistic) equation (map): f:VV: F(x) = rx(1-x), with , which is the case we briefly examined in p. 59. The phase portrait of the orbits of the system on the diagonal is shown in the following figure 1:

Fig.1. Chaotic map. Probability Density Distribution of Periodic orbits The Fr(x):VV map in fig.1, shows that there are a great number of periodic points of various periods, which constitute the set Pern(F) of all periodic points of the map. The set Pern(Fr) (of the total map Fr:VV) consists of points (of period n) in V (see below for explanation). We now analyze the above Fr(x):VV map into its constituents iterate functions has two

etc. We first see that the

fixed points (the one at the origin) of period 1.

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CHAOTIC MAPPING

Fig.3. The graph of

of the mapping: .

has 4 humps and at least 8

fixed points, four of them of

period 4. The rest of its fixed points are carried through the

. From fig.1-3 we derive a two fold general rule: i) The graph of has exactly humps in V and this implies that the graph of has at least fixed points, or equivalently, ii) the Pern(Fr) (of the total map Fr:VV) consists of periodic points of period n in V. The last proposition ii is the result of: a) the fact that the periodic points of a (quadratic) chaotic map are dense in V, i.e. they are eventually fixed in V and b) the property of periodic and fixed points referred above in the section ―fixed points‖ (see also below the ―chaotic regularity 6‖).

219

CHAOTIC MAPPING But the surface values of the fractal constitute an upper bounded and monotonic series and thus it converges to a (non-infinite) limit (despite its infinite length!). The Four Routes to Chaos We have in this book examined in details two closely interrelated routes to chaos, namely: i) the ―period doubling bifurcations for families of maps‖ (see p.52-54), introduced by M.J. Feigenbaum in 1978 and ii) the chaotic mapping Fr(x):VV of the function Fr(x)= rx(1-x), with , both of which constitute the Discrete Logistic model. The chaotic map, characterized by periodic sequences interrupted by a-periodic ones, is also called Pomeau-Manneville model, or Intermittence (Lorenz, Berge, Schuster). An analogous model is the route to chaos through the critical circle map. The third iii) well studied route to chaos is the Appearance of Strange Attractors (D. Rouelle and F. Takens). The fourth is the Hopf bifurcation sequence (p. 262). Lastly, there are many other routes to chaos (see: Eckmann J-P. Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53(1981):643-654). Nevertheless, all chaotic systems share some common features which are as follows: CHAOS: Let V be a set. The iterating mapping F: V V is said to be chaotic on V if: 1. 2. 3. 4. 5.

F has sensitive dependence on initial conditions F is topologically transitive Periodic points of F are dense in V F has a dense orbit in V V is a perfect set

To summarize a chaotic map possesses three features: i) unpredictability, due to its sensitive dependence on the initial conditions, ii) indecomposability due to its been topologically transitive, ii) and an element of regularity, namely its periodic points are dense in it. Therefore in the Chaotic map of fig.2, the orbits are ―disappeared‖, in the sense that they cannot be computed, or predicted whatsoever. This implies that the Chaotic curve of fig.2 is actually a Probability Distribution Function of periodic points of the chaotic region (set V). The chaotic map V is called a ―strange attractor‖.

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DIMITRIOS ANTONIOU

And we again say that the mapping Fr(x) = rx(1-x), with , is chaotic, strange attractor, showing all the above features of all chaotic systems, plus some very individual, such as it been topologically conjugate to its shift and fractal, i.e. it shows scale-free self-similarity. Some chaotic maps possess a special form of sensitive dependence called expansiveness. Expansiveness: The iteration map f: JJ is expansive if there exists ν>0, such that, for any x.y ε J, there exists n such that . Expansiveness differs from sensitive dependence on initial conditions, in that all nearby points eventually separate by at least ν. Elementaries of Circle Map: i) Let

the unit circle in the plane.

We denote a point in by its angle θ in radians, which (the point) is thus determined by any angle of the form θ+2kπ, for an integer k. Now we iterate the function f(θ) = 2θ, in the standard mode: θn+1 = f(θn). Note that f(θ+2π)= f(θ) on the circle. Now, under iteration we take: if

. So that any θ is periodic point of period n if and only , for some integer k, i.e. if and only if , where

is an integer. Hence the periodic

points of period n for f are the

roots of unity. It follows

that the set of periodic points ate dense in . More generally: ii) Translation of the circle. Let iterate the initial angle (point) θo on a circle, in the standard manner:θn+1=f(θn). We then take: . The average increment per iteration of θn, is called ―dressed winding number-λ‖ defined as the limit

of:

If the f is, say, of non-linearity degree 0<K<1 the maps f(θ) behave quite differently depending upon the rationality or irrationality of w. If w= p/q is rational (and p and q are integers), then

230

DIMITRIOS ANTONIOU The significance of Period 3 - Sarkovskii`s Theorem:

Fig.4. The graph and phase portrait of the function f(x)= r x(1-x), with r = 3.839. Fixed point of prime period 3 and periodic points a and b of period 3 and 4 respectively. Another fixed point on the origin. In fig. 4 we depict the graph and phase portrait of the function F(x) 3.839 x(1-x). It shows a fixed point and periodic points a and b of period 3 and 4 respectively. We thus define: Sarkovskii`s Theorem: We look at fig. 4 again. Suppose F has a periodic point of period three. Then f has periodic points of all other periods. But in fig.4 we see only two fixed points and a few p periodic points of periods 3,4.

232

THE DEL OPERATOR II) Analogous Del operations: i) Scalar multiplication: For f(x,y,z):

as defined above and a function

. This is the gradient of f and is a vector. ii) Dot products: For

as defined above and

. This is the divergence of the vector field v and is a scalar quantity. iii) Cross products: For

and v as defined above:

. This is the curl of the vector field v and is a vector. The vector measures local variations in a function and points to the direction of greatest steepness, The scalar quantity measures the tendency of a vector field to represent the divergence (departure) of a fluid. The vector (called the curl of v) depicts a magnitude and axis of rotation, as of fluid in vortex. In biological systems divergence plays the most important role of all the above magnitudes (see next figure).

235

DIMITRIOS ANTONIOU

Fig.1. Several examples of vector fields depicting properties such as divergence, Curl (rotation) of particles. The Laplacian Operator in

236

AUTOCORRELATION FUNCTION Autocorrelation of fractal functions We define the following terms:

A measure of the autocorrelation between f at times separated by h is provided by the autocorrelation function of f:

If there is no correlation, C(h)=0. With C(h) in the last form we can infer a plausible relationship between the autocorrelation function of f and the dimension of graph f. It is proven then that if the autocorrelation function C(h) of a function f satisfies the following power law condition, then the dimension of f equals s.

237

DIMITRIOS ANTONIOU BASIC STATISTICS: I. Real Probability Distribution Functions

Fig. 1. Probability function P(x) (discrete form), or Probability Density function P(x) dx (continuous form). Let plot the variable values x against their frequency appearance (or probability) P(x) (see fig.1). The respective graph of fig.1 is the Probability function P(x) in the discrete form, or the Probability Density function P(x) in the continuous form.

In the continuous form we have: We define as the â&#x20AC;&#x2022;mean valueâ&#x20AC;&#x2013; of the

as follows:

Discrete form: , or number of measurements of x-s.

, where N is the

Continuous form:

If :

then the P(x) (or the P(x)dx)) is normalized.

242

BASIC STATISTICS

Η standard deviation of P(x) or P(x) dx is:

, or

and indicates the degree of dispersion of the distribution. ΙΙ. Complex Probability Distribution Functions

Fig.2. Probability density function (distribution) of a complex function Ψ. Ψ* is the is the complex conjugate of Ψ The Probability density function (distribution) of a complex function Ψ, is:

where Ψ* is the complex conjugate of Ψ The mean value is:

and

Τhe integral Ψ*.

is the convolution of Ψ and

The standard deviation of the distribution is:

243

THE POINCARE-BENDIXSON THEORY THE POINCARE-BENDIXSON THEORY The Poincare-Bendixson theory applies to non-conservative systems and determines the required conditions for the existence of limit-cycles in the plane. One important feature of this theory is the fact that a simple closed curve (f.e. a circle) subdivides a plane into two disjoined open regions (the ―inside‖ and the ―outside‖). This result, known as the Jordan curve theorem (see p. 195, 298), implies that there are restrictions on the trajectories of a smooth two-dimensional phase flow. Summary of the Poincare-Bendixson limit-cycle theorems: If you can find a region in the xy plane containing a single repelling steady state (i.e. unstable node or spiral) and show that the arrows along the boundary D of the region never point outwards, you may conclude that there must be at least one closed periodic trajectory (limit-cycle) inside the region. This is depicted in the following fig.1.

Fig.1. The Poincare-Bendixson theory prescribes the existence of a limitcycle in the xy plane in two equivalent cases: a) Flow cannot leave some

259

DIMITRIOS ANTONIOU THE HOPF BIFURCATION The Hopf Bifurcation theory is another diagnostic tool in establishing the existence of a limit cycle in non-linear and non-conservative systems and importantly it applies, not only for the flow in the plane-as the PoincareBendixson theorem does- but also to higher dimension systems (systems of equations more than two). In summary the Hopf theorem predicts the appearance of a limit cycle about any steady state than undergoes a transition from a stable to an unstable focus (steady state) as some parameter is varied. The result is local in the following sense: a) The theorem only holds for parameter values close to the bifurcation value (the value at which the above transition occurs), 2) The predicted limit cycle is close to the steady state (has a small diameter), 3) The Hopf bifurcation does not specify what happens as the bifurcation parameter is further varied beyond the immediate vicinity of its critical bifurcation value. In the following examples and figures we assume a non-linear oscillating system, with complex eigenvalues λ.

Figure 2: Supercritical Andronov-Hopf bifurcation in the plane. Figure 1 illustrates what happens close to a parameter bifurcation critical value β=0 in the plane. We observe that as the parameter β<0 increases from its initial stable steady state (stable spiral) to its higher value β*=0 in which a structurally unstable neutral center is created, which is encircled by infinitely many stable concentric closed orbits. As the β* further increases, there is a range of β values such that 0<β*<β=c, for which only a single closed orbit (a limit cycle) with a very small diameter is emerged which surrounds the evolved new unstable center. As β* further increases to higher values β=c>β*, the

262

THE HOPF BIFURCATION

diameter of the limit cycle increases in proportion to , surrounding now the final unstable steady state and β=c. Because the limit cycle appears and exists for values β above β*, this phenomenon is called the supercritical bifurcation.

Figure 3: Subcritical Andronov-Hopf bifurcation in the plane. The inverse phenomenon also occurs. Namely figure 3 shows that an unstable limit cycle (*) accompanies the stable steady state until β<0 increases to β=β* , in which the limit cycle disappears and a neutral center is created. For higher values of β*<β the system ends at β=c with creation an unstable steady state. (*) the emergence and sustainability of such an (unstable) limit cycle around a stable steady state, has been originated by another nearby unstable steady state (see below).

Figure 4: Supercritical Hopf bifurcation in the 3D-space. 263

LEVEL CURVES LEVEL (CONTOUR) CURVES Level (contour) curves can be conceived geometrically as solution curves with constant value, of surface-functions z = f(x,y), or informally, as the constant-altitude contours on a topographic map of a mountain range. Each level curve corresponds to a fixed â&#x20AC;&#x2022;total energyâ&#x20AC;&#x2013;, a fixed potential, or a fixed height on the mountain, depending on the way one interprets the conserved quantity. Equivalently, level surfaces are iso-thermic surfaces of 3-dimensional systems f(x,y,z) (see fig.5). Among them, of particular importance are the neutrally stable (and structurally unstable) level curvescycles of conservative autonomous systems. An important property of the level curve phase portrait of any solution curve (function) f(x,y), is that the gradient vectors of it are always perpendicular (normal) to all its points. The set of vectors of the level curves phase portrait, constitutes the vector gradient field of the phase portrait of f(x,y) (see also p. 194 fig.4).

Fig.5a. Circular Level curves of a conic surface

267

LEVEL CURVES

Fig.5d,e,f.. Parabolic Level curves of a elliptic paraboloid surface

269

DIMITRIOS ANTONIOU

fibonacci

Fig.5g (1-8). Spiral curves of spiroid surfaces

270

CLASSICAL LOTKA-VOLTERRA MODEL The Classical Lotka-Volterra Model (Predator-Prey Model) The mathematical and physical features of the limit cycles are governed by the Poincare-Bendixson and Hopf bifurcation theories and encompass oscillating phenomena in Biology, such as the modified Lotka-Volterra systems (see next), in Chemistry, such as the Belousov-Zhabotinsky chemical reaction (p.282) and finally in physiology, such as the Circadian rhythms (p.300). We`ll briefly examined next the oscillating Biological and Chemical systems. Introduction: A) The classical conservative Lotka-Volterra model The classical Lotka-Voltera system (model) (p. 165) consists of the two following equations:

(1a,b) Then the following two possible steady states of the system can exist:

The stability properties are given in pages 192-194.

The Jacobian matrix (*) of the system is: and for steady state

is:

whereas for the steady state

is:

Thus the respective eigenvalues are:

. Thus the

is a saddle point.

273

DIMITRIOS ANTONIOU

. Thus the

is a neutral center.

Fig.6. The phase-plane of the classical Lotka-Volterra model We already know from p.193 that Lotka-Volterra systems form neutral center near their steady states

, since:

and Î˛ =0. The real part r of the eigenvalue

zero and the amplitude of the solution x(t) does not change. The amplitude depends on the initial population level, a property shared by all conservative systems.

274

DIMITRIOS ANTONIOU

Fig.7. The phase-plane of the classical Lotka-Volterra model combined with a logistic growth of the prey. The steady state becomes a stable spiral with somewhat depressed predator population levels. C) The non-conservative modified classical Lotka-Volterra Model: Kolmogorov`s Theorem- Oscillations (limit cycles) in Biology Because non-linear continuous conservative biological oscillating systems which form stable neutral centers â&#x20AC;&#x201C;such as the non-linear Lotka-Volterraare structurally unstable (see p. 171), we look for a modified nonconservative version of the Lotka-Volterra â&#x20AC;&#x201C; introducing further assumptions- which would ensure the existence of structurally stable limit cycles according to Poincare-Bendixson or Hopf`s bifurcation theories. And indeed such models have been modelled, whose principles are included in the following famous Kolmogorov`s theorem, which is summarized as follows: Consider the classical Lotka-Volterra (see p. 165) conservative and structurally unstable model: dx/dt = xf(x.y)

(2a)

dy/dt = yg(x,y)

(2b)

276

DIMITRIOS ANTONIOU

Fig.6: Modified non-conservative Lotka-Volterra model. (a): It expresses Kolmogorov`s assumptions 1-4 and ensures that far away flow is directed towards decreasing x and y values, (b): It expresses the assumption 5-6 , Flow along x can be reversed by allowing for a steady state f( , 0) =0. in the absence of predators, (c) It expresses the assumption 8-10. Flow along the y axis can be controlled by bending the y

280

DIMITRIOS ANTONIOU PARTIAL DERIVATIVES Basic definition In mathematics a partial derivative of a function of several variables, is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The first order partial derivative of a function f (x,y) with respect to the variable x is variously denoted by:

. whereas the:

, is the partial derivative of f(x,y) with respect to y. The partial derivative symbol Î¸ is a rounded letter, derived but distinct from the Greek letter delta Î´. Formal definition

Let z = f(x,y) be a function of two variables. Then we define the partial derivatives as: Definition of Partial Derivatives:

if these limits exist. Algebraically, we can think of the partial derivative of a function z= f(x,y) with respect to x as the derivative of the function with y held

284

PARTIAL DERIVATIVES

Fig.2. Both tangents T1 and T2 of fig.1 define a plane tangent to P(a,b,c) 2) Suppose that Ć&#x2019; is a function of more than one variable. For instance,

Fig. 3. A graph of z = x2 + xy + y2. For the partial derivative at (1, 1, 3) that leaves y constant, the corresponding tangent line is parallel to the xz-plane.

289

PARTIAL DERIVATIVES

Fig.5. For the graph of a function (blue) the partial derivative can be thought of as taking a plane (purple), finding the curve that the plane contains (yellow), then taking the derivative of this curve. Conclusion: The partial derivative in three dimensions can be thought of as taking a "slice" of a function from and finding the derivative of the curve within this slice. Taking a slice means we take a plane along which one of the input variables is constant, find the curve contained by this plane, then differentiate this curve with respect to the other input variable.

291

MATRICES MATRICES Definition-I A matrix is a reactangular arrangement of numbers. For example:

An alternative notation uses large parentheses instead of box brackets:

The horizontal and vertical lines in a matrix are called rows and columns, respectively. The numbers in the matrix are called its entries or its elements. To specify a matrix's size, a matrix with m rows and n columns is called an m-by-n matrix or m × n matrix, while m and n are called its dimensions. The above is a 4-by-3 matrix. A matrix with one row (an 1 × n matrix) is called a row vector, and a matrix with one column (an m × 1 matrix) is called a collumn vector. Any row or column of a matrix determines a row or column vector, obtained by removing all other rows or columns respectively from the matrix. For example, the row vector for the third row of the above matrix A is

. When a row or column of a matrix is interpreted as a value, this refers to the corresponding row or column vector. For instance one may say that two different rows of a matrix are equal, meaning they determine the same row vector. In some cases the value of a row or column should be interpreted just as a sequence of values (elements of R n if entries are real numbers) rather than as a matrix. For instance, when

305

MATRIX (LINEAR) TRANSFORMATIONS Linear transformations Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A real mby-n matrix A gives rise to a linear transformation Rn → Rm mapping each vector x in Rn to the (matrix) product Ax, which is a vector in Rm. Conversely, each linear transformation f: Rn → Rm arises from a unique m-by-n matrix A: Let consider the rectangle S in the uυ plane, whose origin is (xo,yo), its dimensions are Δu and Δυ and which is transformed by the transforming function T to region R in the xy plane: T: S  R, or R = T(S).

The image of S is a region R in the xy plane, whose the boundary point of origin is

319

MATRIX (LINEAR) TRANSFORMATIONS

Rotation by π/6R = 30°

Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps: if a k-by-m matrix B represents another linear map g : Rm → Rk, then the composition g . f is represented by BA since (g . f)(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x. The last equality follows from the above-mentioned associativity of matrix multiplication.

323

THE JACOBIAN MATRIX The Jacobian Matrix We must distinguish between the Jacobian matrix and the determinant-det of it, which is called â&#x20AC;&#x2022;Jacobianâ&#x20AC;&#x2013; as follows: In considering the following system of two differential equations we have:

329

EIGENVECTORS Computation of eigenvalues, and the characteristic equation Let the system of two differential equations:

We write it with the vector notation :

where Ax notes matrix multiplication, is a vector whose entries are dx/dt and dy/dt and x(x(t),y(t)) is the position vector, which represents the solution curve x(t) or the (could be) eigenvector x of the system (see p. 342). When a transformation is represented by a square matrix A, the eigenvalue equation can be expressed as: (note that x cannot be â&#x20AC;&#x2022;cancelledâ&#x20AC;&#x2013; in both sides of the equation, since Av stands for matrix myltiplication). However we can also write: , where I is the identity matrix: Ix = x.

343

EIGENVECTORS

Now we can have a more profound insight of the phase portrait of linear systems or of the non-linear ones near their steady state, described in p. 191 fig.3, with real eigenvalues 位i (i=1,2).

347

DIMITRIOS ANTONIOU If the eigenvalues λi (i=1,2) are complex, then the eigenvectors and solution curves are oscillating and take the form of the d,e,f flow patterns of the figure 4 of p.194:

Fig.5 (fig.4 of p. 194). General flow behavior (solution curves) of linear and local behavior of non-linear systems close to the steady states of the latter . The flow patterns d,e,f are for complex eigenvalues λi (i=1,2). The β γ parameter plane showing the 6 regions in which the steady state is an unstable (a) or stable (c) node, an unstable (d) or stable (f) spiral, or a saddle point (b,b), according to the values of β and γ. Finally the set (e) of disjoint neutrally stable concentric closed curves (trajectories) encircling the neutral stable center (origin) is structurally unstable, in the sense that small changes may disrupt its balance (steadiness) (p.197). The a-e solution curves could also be level (contour) flow curves of a surface flow of a conservative system z = f(x,y) (see p.267 - 282). In this case, we can geometrically construct the real surface flow (object).

350

VECTOR CALCULUS

Euclidean vectors

A vector going from A to B In elementary mathematics, physics and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or - as here simply a vector) is a geometric object that has both a magnitude (or length) and direction. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by

. A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "one who carries". The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, abstraction, multiplication, , and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutatinity, associativity and distributivity.. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vectro space.. Vectors play an important role in physicsâ&#x20AC;&#x2013; velocity and accelearation of a moving object and forces acting on it are all described by vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (such as position or displacement), their magnitude and direction can be still represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system. used to describe it. Other vector-like objects that describe physical

355

VECTOR CALCULUS

In introductory physics classes, these three special vectors are often instead denoted ,

the versors of the three dimensional space, in which the hat symbol (^) typically denotes unit vectors (vectors with unit length). The notation ei is compatible with the index notation and the summation convention and the commonly used in higher level mathematics, physics, and engineering. The use of Cartesian versors such as as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of cylindrical unit vectors

or spherical unit vectors

. The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry respectively. Basic properties The following section uses the Cartesian coordinate system, with basis vectors

and assume that all vectors have the origin as a common base point. Such vectors are called ―position vectros‖, while a ―position vector‖ a will be written as:

Position vector, displacement vector The position of a point x=(x1, x2, x3) in three dimensional space can be represented as a position vector whose base point is the origin

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