01050412

Page 1

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016.

The Bifurcation of Stage Structured Prey-Predator Food Chain Model with Refuge Azhar Abbas Majeed and Noor Hassan Ali Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq ABSTRACT :In this paper, we established the condition of the occurrence of local bifurcation (such as saddlenode, transcritical and pitchfork)with particular emphasis on the hopf bifurcation near of the positive equilibrium point of ecological mathematical model consisting of prey-predator model involving prey refuge with two different function response are established. After the study and analysis, of the observed incidence transcritical bifurcation near equilibrium point đ??¸0 , đ??¸1 , đ??¸2 as well as the occurrence of saddle-node bifurcation at equilibrium point đ??¸3 .It is worth mentioning, there are no possibility occurrence of the pitch fork bifurcation at each point. Finally, some numerical simulation are used to illustration the occurrence of local bifurcation o this model. KEYWORDS -ecological model, Equilibrium point, Local bifurcation, Hopf bifurcation.

I.

INTRODUCTION

A bifurcation mean there is a change in the stability of equilibria of the system at the value. Many, if not most differential equations depend of parameters. Depending on the value of these parameters, the qualitative behavior of systems solution can be quite different [1]. Bifurcation theory studies the qualitative changes in the phase portrait, for example, the appearance and disappearance of equilibria, periodic orbits, or more complicated features such as strange attractors. The methods and results of bifurcation theory are fundamental to an understanding of nonlinear dynamical systems. The bifurcationis divided into two principal classes: local bifurcations and global bifurcations. Local bifurcations, which can be analyzed entirely through changes in the local stability properties of equilibria, periodic orbit or other invariant sets as parameters cross through critical thresholds such as saddle node, transcritical, pitchfork, period-doubling (flip), Hopf and Neimark (secondary Hopf) bifurcation. Global bifurcations occur when larger invariant sets, such as periodic orbits, collide with equilibria. This causes changes in the topology of the trajectories in phase space which cannot be confined to a small neighborhood, as is the case with local bifurcations. In fact, the changes in topology extend out of an arbitrarily large distance (hence “global�). Such as homoclinic in which a limit cycle collides with a saddle point, and heteroclinic bifurcation in which a limit cycle collides with two or more saddle points, and infinite-periodic bifurcation in which a stable node and saddle point simultaneously occur on a limit cycle [2]. the name "bifurcation" in 1885 in the first paper in mathematics showing such a behavior also later named various types of stationary points and classified them. Perko L. [3] established the conditions of the occurrence of local bifurcation (such as saddle-node, transcritical and pitchfork). However,the necessary condition for the occurrence

www.ijmret.org

of the Hopf bifurcation presented by Hirsch M.W. and Smale S. [4 ]. In this chapter, we will establish the condition of the occurrence of local bifurcation and Hopf bifurcation around each of the equilibrium point of a mathematical model proposed by AzharM,and Noor Hassan [ 5 ].

II.

Model formulation [ 5 ]:

Anecological mathematical model consisting of prey-predator model involving prey refuge with two different function response is proposed and analyzed in [ 5 ]. dW1 W2 = Îą W2 1 − − β W1 dT k dW2 = β W1 − d1 W2 − c1 1 − m W2 ( 1 ) dT dW3 c2 W3 = e1 c1 1 − m W2 W3 − d2 W3 − W dT b + W3 4 dW4 e2 c2 W3 = W − d3 W4 dT b + W3 4 with initial conditions Wi ( 0 ) ≼ 0 , i = 1 , 2 , 3 , 4 . Not that the above proposed model has twelve parameters in all which make the analysis difficult. So in order to simplify the system, the number of parameters is reduced by using the following dimensionless variables and parameters : β d1 d2 t = Îą T , a1 = , a 2 = , a3 = , Îą Îą Îą e1 c1 k c2 b c1 a4 = , a5 = , a6 = , Îą Îą Îą e2 c2 d3 W1 W2 a7 = , a8 = , w1 = , w2 = , Îą Îą k k c1 W3 c1 W4 w3 = , w4 = . Îą Îą Then the non-dimensional from of system (1) can be written as: dw1 w2 1 − w2 = w1 − a1 dt w1 = w1 f1 w1 , w2 , w3 , w4

Page 4

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. dw2 a1 w1 = w2 − a 2 − 1 − m w3 dt w2 = w2 f2 w1 , w2 , w3 , w4 (2) dw3 a 5 w4 = w3 a 4 1 − m w2 − a 3 − dt a 6 + w3 = w3 f3 w1 , w2 , w3 , w4 dw4 a 7 w3 = w4 − a 8 = w4 f4 w1 , w2 , w3 , w4 dt a 6 + w3 with w1 0 ≼ 0 , w2 0 ≼ 0 , w3 0 ≼ 0 and w4 0 ≼ 0 . It is observed that the number of parameters have been reduced from twelve in the system 1 to nine in the system 2 . Obviously the interaction functions of the system 2 are continuous and have continuous partial derivatives on the following positive four dimensional space. w1 , w2 , w3 , w4 ∈ R4 âˆś w1 0 ≼ 0 , R4+ = w2 0 ≼ 0 , w3 0 ≼ 0 w4 0 ≼ 0. Therefore these functions are Lipschitzian on R4+ , and hence the solution of the system 2 exists and is unique. Further, all the solutions of system 2 with non-negative initial conditions are uniformly bounded as shown in the following theorem. Theorem 1 : All the solutions of system 2 which initiate in R4+ are uniformly bounded.

III.

The stability Analysis of Equilibrium Points of system (2) [5]

It is observed that, system (2) has at most four biological feasible equilibrium point which are mentioned with their existence condition in [5 ] as in the following: 1.the Vanishing Equilibrium point: E0 = (0,0,0,0) always exists and E0 is locally asymptotically stable in Int R4+ if the following condition hold a2 > 1 (3. a) However, it is (a saddle point) unstable otherwise. More details see [ 5 ] 2.The freepredators equilibrium point đ?‘Źđ?&#x;? = đ?’˜đ?&#x;? , đ?’˜đ?&#x;? , đ?&#x;Ž , đ?&#x;Ž exists uniquely in đ??źđ?‘›đ?‘Ą. đ?‘…+2 if and only if the following condition hold: đ?‘Ž3 đ?‘¤2 < (3. đ?‘?) đ?‘Ž4 (1 − đ?‘š) The following second order polynomial equation đ?œ†2 + đ??´1 đ?œ† + đ??´2 = 0 , which gives the other two eigenvalues of đ??˝ đ??¸0 by: −đ??´1 1 đ?œ†0đ?‘¤ 1 = + đ??´1 2 − 4 đ??´2 , 2 2 −đ??´1 1 đ?œ†0đ?‘¤ 2 = − đ??´1 2 − 4 đ??´2 . 2 2 đ??¸0 is locally asymptotically stable in the đ?‘…+4 . However, it is a saddle point otherwise .

www.ijmret.org

3.The freetoppredator equilibrium pointđ??¸2 = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 ,0 exists uniquely in the đ??źđ?‘›đ?‘Ą. đ?‘…+3if the following condition are holds: đ?‘Ž3 < min đ?‘Ž4 1 − đ?‘š , đ?‘Ž4 1 − đ?‘š 1 − đ?‘Ž2 . 3đ?‘‘ The following three order polynomial equation đ?‘Ž7 đ?‘¤3 đ?œ†3 + đ?‘…1 đ?œ†2 + đ?‘…2 đ?œ† + đ?‘…3 −đ?‘Ž8 + −đ?œ† đ?‘Ž6 + đ?‘¤3 =0 where đ?‘…1 = −( đ?‘?11 + đ?‘?22 ) > 0 , đ?‘…2 = − đ?‘?11 đ?‘?22 + đ?‘?23 đ?‘?32 + đ?‘?12 đ?‘?21 , đ?‘…3 = đ?‘?11 đ?‘?23 đ?‘?32 > 0, đ?‘Ž đ?‘¤ So, either−đ?‘Ž8 + đ?‘Ž 7+ đ?‘¤3 − đ?œ† = 0 . 6

3

Orđ?œ†3 + đ?‘…1 đ?œ†2 + đ?‘…2 đ?œ† + đ?‘…3 = 0 . Hence from equation 2.9 đ?‘? we obtain that: đ?‘Ž7 đ?‘¤3 đ?œ†2đ?‘¤ 3 = −đ?‘Ž8 + , đ?‘Ž6 + đ?‘¤3 which is negative if inaddition of condition (2. 5 h)in [ ] the following condition hold: đ?‘Ž6 đ?‘Ž8 đ?‘¤3 < . 3đ?‘’ đ?‘Ž7 − đ?‘Ž8 On the other hand by using Routh-Hawirtiz criterion equation in [5] has roots ( eigenvalues ) with negative real parts if and only if đ?‘…1 > 0 , đ?‘…3 > 0 đ?‘Žđ?‘›đ?‘‘ ∆ = đ?‘…1 đ?‘…2 − đ?‘…3 > 0 Straightforword computation shows that ∆ > 0 đ?‘Ž3 1 provided that > (3đ?‘“) đ?‘Ž4 1 − đ?‘š 2 Now it is easy to verify that đ?‘…1 > 0 andđ?‘…3 > 0 under condition 2.4 đ?‘” .Then all the eigenvalues đ?œ†đ?‘¤ 1 , đ?œ†đ?‘¤ 2 and đ?œ†đ?‘¤ 3 of equation 2.9 đ?‘‘ have negative real parts. So, đ??¸2 is locally 4. Finally, the Positive (Coexistence)Equilibrium point: đ??¸3 = đ?‘¤1∗ , đ?‘¤2∗ , đ?‘¤3∗ , đ?‘¤4∗ exists and it is locally asymptotically stable, as shown in [ ].

IV.

The local bifurcation analysis of system đ?&#x;?

In this section, the effect of varying the parameter values on the dynamical behavior of the system 2 around each equilibrium points is studied. Recall that the existence of non hyperbolic equilibrium point of system 2 is the necessary but not sufficient condition for bifurcation to occur. Therefore, in the following theorems an application to the Sotomayor’s theorem for local bifurcation is appropriate. Now, according to Jacobian matrix of system 2 given in equation 2.7 , it is clear to verify that for any nonzero vector V= v1 , v2 , v3 , v4 T we have:

Page 5

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. 0 0 0 0 0 −1 0 0 , đ??ˇđ?‘“đ?‘Ž 2 đ?‘Š , đ?‘Ž2 = 0 0 0 0 0 0 0 0 where đ??ˇđ?‘“đ?‘Ž 2 đ?‘Š , đ?‘Ž2 represents the derivative of đ?‘“đ?‘Ž 2 đ?‘Š, đ?‘Ž2 with respect to đ?‘Š = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 , đ?‘¤4 đ?‘‡ . Further ,it is observed that

đ??ˇ2 đ?‘“ đ?‘‰ , đ?‘‰ = −2 đ?‘Ł22 −2 1 − đ?‘š đ?‘Ł2 đ?‘Ł3 2 đ?‘Ž4 1 − đ?‘š đ?‘Ł2 đ?‘Ł3 + 2 2

đ?‘Ž 5 đ?‘Ž 6 đ?‘Ł3

�3 � 4

đ?‘Ž 6 +đ?‘¤ 3 2 đ?‘Ž 6 +đ?‘¤ 3 đ?‘Ž 6 đ?‘Ž 7 đ?‘Ł3 đ?‘Ł đ?‘¤ đ?‘Ł4 − 3 4 đ?‘Ž +đ?‘¤ 2 đ?‘Ž +đ?‘¤ 6

3

6

− đ?‘Ł4

3

(3.1) 0 0 andđ??ˇ3 đ?‘“ đ?‘‰ , đ?‘‰ , đ?‘‰ =

−2 6

đ?‘Ž 5 đ?‘Ž 6 đ?‘Ł32

3�3 � 4 � 6 +� 3 3 � 6 +� 3 � 6 � 7 �32 �3 � 4 � 6 +� 3 3 � 6 +� 3

đ??ˇđ?‘“đ?‘Ž 2 đ??¸0 , đ?‘Ž2∗ đ?‘‰ 0 + đ?‘Ł4

.

− đ?‘Ł4

In the following theorems the local bifurcation conditions near the equilibrium point are established. Theorem (3.1): If the parameter đ?‘Ž2 passes through the value đ?‘Ž2∗ = 1 , then the vanishing equilibrium point đ??¸0 transforms into non-hyperbolic equilibrium point and system (2.2) possesses a transcritical bifurcation but no saddle-node bifurcation, nor pitchfork bifurcation can occur at đ??¸0 . . Proof: According to the Jacobian matrix đ??˝ đ??¸0 given by Eq. 2.8 đ?‘Ž the system 2 at the equilibrium point đ??¸0 has zero eigenvalue (say đ?œ†0đ?‘Ľ = 0 ) atđ?‘Ž2 = đ?‘Ž2∗ , and the Jacobian matrixđ??˝0 with đ?‘Ž2 = đ?‘Ž2∗ becomes: −đ?‘Ž1 1 0 0 đ?‘Ž1 −1 0 0 ∗ ∗ đ??˝0 = đ??˝ đ?‘Ž2 = đ?‘Ž2 = −đ?‘Ž3 0 0 0 0 −đ?‘Ž6 0 0 đ?‘‡

Now, let đ?‘‰ 0 = đ?‘Ł10 , đ?‘Ł20 , đ?‘Ł30 , đ?‘Ł40 be the eigenvector corresponding to the eigenvalueđ?œ†0đ?‘Ľ = 0 .Thus đ??˝0∗ − đ?œ†0đ?‘¤ 1 đ??ź đ?‘‰ 0 = 0 , which gives: đ?‘Ł20 = đ?‘Ž1 đ?‘Ł10 , đ?‘Ł30 = 0 , đ?‘Ł40 = 0 and đ?‘Ł10 any nonzero real number. 0

0

0

0

�

Let đ?›š 0 = đ?œ“1 , đ?œ“2 , đ?œ“3 , đ?œ“4 be the eigenvector associated with the eigenvalue đ?œ†0đ?‘¤ 1 = 0 of the matrix đ??˝0∗đ?‘‡ . Then we have, đ??˝0∗đ?‘‡ − đ?œ†0đ?‘¤ 1 đ??ź đ?›š 0 = 0 . By solving this equation for đ?›š0 we obtain, đ?‘‡

đ?›š 0 = đ?œ“10 , đ?œ“10 , 0 , 0 , where đ?œ“10 any nonzero real number. Now, consider: đ?œ•đ?‘“ đ?œ•đ?‘“1 đ?œ•đ?‘“2 đ?œ•đ?‘“3 đ?œ•đ?‘“4 đ?‘‡ = đ?‘“đ?‘Ž 2 đ?‘Š , đ?‘Ž2 = , , , đ?œ•đ?‘Ž2 đ?œ•đ?‘Ž2 đ?œ•đ?‘Ž2 đ?œ•đ?‘Ž2 đ?œ•đ?‘Ž2 đ?‘‡ = 0 , −đ?‘¤2 , 0 , 0 . So, đ?‘“đ?‘Ž 2 đ??¸0 , đ?‘Ž2∗ = 0 , 0 , 0 , 0 đ?‘‡ and đ?‘‡

hence đ?›š 0 đ?‘“đ?‘Ž 2 đ??¸0 , đ?‘Ž2∗ = 0 . Therefore, according to Sotomayor’s theorem the saddle-node bifurcation cannot occur. While the first condition of transcritical bifurcation is satisfied. Now, since

www.ijmret.org

0

đ?‘Ł1

0 0 0 0 0 −1 0 0 = 0 0 0 0 0 0 0 0

đ?‘Ž1 đ?‘Ł10 0 0

0 0 −đ?‘Ž 1 đ?‘Ł1 = , 0 0 đ?‘‡ đ?›š0 đ??ˇđ?‘“đ?‘Ž 2 đ??¸0 , đ?‘Ž2∗ đ?‘‰ 0 = −đ?‘Ž1 đ?‘Ł10 đ?œ“10 ≠0 . Now, by substitutingđ?‘‰ 0 in (3.1) we get: 0

−2 đ?‘Ž12 đ?‘Ł1 đ??ˇ2 đ?‘“ đ??¸0 , đ?‘Ž2∗ đ?‘‰ 0 , đ?‘‰

0

=

.

0 0 0

Hence, it is obtain that: đ?‘‡ đ?›š 0 đ??ˇ2 đ?‘“ đ??¸0 , đ?‘Ž2∗ đ?‘‰ 0 , đ?‘‰ 0

2

2

= −2 đ?‘Ž12 đ?œ“10 đ?‘Ł10 ≠0 . Thus, according to Sotomayor’s theorem (1.15) system 2 has transcrirtical bifurcation at đ??¸0 with the parameterđ?‘Ž2 = đ?‘Ž2∗ . Theorem;If the parameter đ?‘Ž3 passes through the value đ?‘Ž3∗ = đ?‘Ž4 1 − đ?‘š đ?‘¤2 , then the free predatorsequilibrium point đ??¸1 = đ?‘¤1 , đ?‘¤2 ,0 ,0 transforms into non-hyperbolic equilibrium point and system (2) possesses a transcritical bifurcation but no saddle-node bifurcation, nor pitchfork bifurcation can occur at đ??¸1 = đ?‘¤1 , đ?‘¤2 ,0 ,0 . . Proof: According to the Jacobian matrixđ??˝1 given by Eq. 2.9 đ?‘Ž the system 2 at the equilibrium pointđ??¸1 has zero eigenvalue (sayđ?œ†1đ?‘¤ 3 = 0) at đ?‘Ž3 = đ?‘Ž3∗ , and the Jacobian matrix đ??˝1 with đ?‘Ž3 = đ?‘Ž3∗ becomes: đ??˝1∗ = đ??˝ đ?‘Ž3 = đ?‘Ž3∗ −đ?‘Ž1 1 − 2đ?‘¤2 0 0 đ?‘Ž1 −đ?‘Ž 2 − 1 − đ?‘š đ?‘¤2 − 1 − đ?‘š đ?‘¤2 = 0 0 0 0 0 −đ?‘Ž + đ?‘Ž 1 − đ?‘š đ?‘¤2 0 0 6 7 đ?‘‡

Let đ?‘‰ 1 = đ?‘Ł11 , đ?‘Ł21 , đ?‘Ł31 , đ?‘Ł41 be the eigenvector corresponding to the eigenvalueđ?œ†1đ?‘¤ 3 = 0 . Thus đ??˝1∗ − đ?œ†1đ?‘¤ 3 đ??ź đ?‘‰ 1 = 0 ,which gives: 1 − 2đ?‘Ž2 1 − đ?‘š 1 đ?‘Ł11 = đ?‘Ł3 , đ?‘Ł21 đ?‘Ž1 = − 1 − đ?‘š đ?‘Ł31 đ?‘Žđ?‘›đ?‘‘ đ?‘Ł41 =0 , 1 where đ?‘Ł3 any nonzero real number.

Page 6

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. đ?‘‡

Let đ?›š 1 = đ?œ“11 , đ?œ“21 , đ?œ“31 , đ?œ“41 be the eigenvector associated with the eigenvalueđ?œ†1đ?‘¤ 3 = 0 of the matrix đ??˝1∗đ?‘‡ . Then we have, ∗đ?‘‡ 1 đ??˝1 − đ?œ†1đ?‘¤ 3 đ??ź đ?›š = 0 . By solving this equation for đ?›š 1

we

đ?›š 1 = 0 , 0 , đ?œ“31 , 0

obtain,

�

,

đ?œ“31

where any nonzero real number. Now, consider: đ?œ•đ?‘“ đ?œ•đ?‘“1 đ?œ•đ?‘“2 đ?œ•đ?‘“3 đ?œ•đ?‘“4 đ?‘‡ = đ?‘“đ?‘Ž 3 đ?‘Š , đ?‘Ž3 = , , , đ?œ•đ?‘Ž3 đ?œ•đ?‘Ž3 đ?œ•đ?‘Ž3 đ?œ•đ?‘Ž3 đ?œ•đ?‘Ž3 đ?‘‡ = 0 , 0 , −đ?‘¤3 , 0 . So, đ?‘“đ?‘Ž 3 đ??¸1 , đ?‘Ž3∗ = 0 , 0 , 0 , 0 đ?‘‡ and đ?‘‡

hence đ?›š 1 đ?‘“đ?‘Ž 3 đ??¸1 , đ?‘Ž3∗ = 0 . Therefore, according to Sotomayor’s theorem (1.15) the saddle-node bifurcation cannot occur. While the first condition of transcritical bifurcation is satisfied. Now, since 0 0 0 0 0 0 0 0 đ??ˇđ?‘“đ?‘Ž 3 đ?‘Š , đ?‘Ž3 = , 0 0 −1 0 0 0 0 0 where đ??ˇđ?‘“đ?‘Ž 3 đ?‘Š , đ?‘Ž3 represents the derivative of đ?‘“đ?‘Ž 3 đ?‘Š , đ?‘Ž3 with respect to đ?‘Š = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 , đ?‘¤4 đ?‘‡ .Further ,it is observed that đ?‘Ł11 0 0 0 0 đ?‘Ł21 đ??ˇđ?‘“đ?‘Ž 3 đ??¸1 , đ?‘Ž3∗ đ?‘‰ 1 = 0 0 0 0 0 0 −1 0 đ?‘Ł31 0 0 0 0 đ?‘Ł41 0 0 = , −đ?‘Ł31 0 đ?‘‡ đ?›š1 đ??ˇđ?‘“đ?‘Ž 3 đ??¸1 , đ?‘Ž3∗ đ?‘‰ 1 = −đ?‘Ł31 đ?œ“31 ≠0 . Now, by substitutingđ?‘‰ 1 in (3.1) we get: đ??ˇ2 đ?‘“ đ??¸1 , đ?‘Ž3∗ đ?‘‰ 1 , đ?‘‰ 1

=

2 1−đ?‘š

2

đ?‘Ł31

2 1−đ?‘š

2

đ?‘Ł31

−2 đ?‘Ž4 1 − đ?‘š

2

0 Hence, it is obtain that: đ?‘‡ đ?›š 1 đ??ˇ2 đ?‘“ đ??¸1 , đ?‘Ž3∗ đ?‘‰ 1 , đ?‘‰ 1 2

đ?œ“31

đ?‘Ł31

2 2

.

đ?‘Ł31

2

Proof: According to the Jacobian matrixđ??˝2 given by Eq. 2.10 đ?‘Ž the system 2.2 at the equilibrium pointđ??¸2 has zero eigenvalue (say đ?œ†2đ?‘¤ 4 = 0) at đ?‘Ž8 = đ?‘Ž8∗ , and the Jacobian matrix đ??˝2 with đ?‘Ž8 = đ?‘Ž8∗ becomes: đ??˝2∗ = đ??˝ đ?‘Ž8 = đ?‘Ž8∗ = đ?‘’đ?‘–đ?‘— where 4Ă—4 đ?‘›11 = −đ?‘Ž1 , đ?‘›12 = 1 − 2đ?‘¤2 , đ?‘›13 = 0 , đ?‘›14 = 0 , đ?‘›21 = đ?‘Ž1 , đ?‘›22 = −đ?‘Ž2 − 1 − đ?‘š đ?‘¤3 , đ?‘›23 = − 1 − đ?‘š đ?‘¤2 , đ?‘›24 = 0, đ?‘›31 = 0 , đ?‘›32 = đ?‘Ž4 1 − đ?‘š đ?‘¤3 , đ?‘›33 = −đ?‘Ž3 + đ?‘Ž4 1 − đ?‘š đ?‘¤2 , đ?‘Ž5 đ?‘¤3 đ?‘›34 = − , đ?‘› = 0 , đ?‘›42 = 0 , đ?‘›43 = 0 , đ?‘›44 đ?‘Ž6 + đ?‘¤3 41 =0 . đ?‘‡

Let đ?‘‰ 2 = đ?‘Ł12 , đ?‘Ł22 , đ?‘Ł32 , đ?‘Ł42 be the eigenvector corresponding to the eigenvalueđ?œ†2đ?‘¤ 3 = 0 .Thus đ??˝2∗ − đ?œ†2đ?‘¤ 3 đ??ź đ?‘‰ 2 = 0 ,which gives: đ?‘Ž5 đ?‘Ž4 1 − đ?‘š − 2đ?‘Ž3 đ?‘Ł12 = đ?‘Ł2 , 2 đ?‘Ž1 đ?‘Ž6 đ?‘Ž4 1 − đ?‘š 2 1 − đ?‘Ž2 − đ?‘Ž1 đ?‘Ž3 đ?‘Ž4 4 đ?‘Ł22 = đ?‘Žđ?‘›đ?‘‘ đ?‘Ł32

đ?‘Ž5 (1 − đ?‘š) đ?‘Ł2 đ?‘Ž4 đ?‘Ž6 1 − đ?‘š + đ?‘Ž4 1 − đ?‘š − đ?‘Ž3 4

đ?‘Ž5 đ?‘Ł2 , đ?‘Ž4 đ?‘Ž6 1 − đ?‘š + đ?‘Ž4 1 − đ?‘š − đ?‘Ž3 4 wheređ?‘Ł42 any nonzero real number . =−

�

Let đ?›š 2 = đ?œ“12 , đ?œ“22 , đ?œ“32 , đ?œ“42 be eigenvector associated with eigenvalue đ?œ†2đ?‘¤ 3 = 0 of the matrix đ??˝2∗đ?‘‡ .Then have, đ??˝2∗đ?‘‡ − đ?œ†2đ?‘¤ 4 đ??ź đ?›š 2 = 0. By solving

the the we this

equation forđ?›š 2 we obtain, đ?›š 2 = 0 , 0 , 0 , đ?œ“42

=

2

Theorem (3.3):If the parameter đ?‘Ž8 passes through đ?‘Ž đ?‘¤ the value đ?‘Ž8∗ = 7 4 , then the free top predators đ?‘Ž 6 +đ?‘¤ 3

equilibrium point đ??¸2 = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 ,0 transforms into non-hyperbolic equilibrium point and system (2.2) possesses a transcritical bifurcation but no

�

,

2 đ?œ“4

where any nonzero real number. Now, consider: đ?œ•đ?‘“ đ?œ•đ?‘“1 đ?œ•đ?‘“2 đ?œ•đ?‘“3 đ?œ•đ?‘“4 đ?‘‡ = đ?‘“đ?‘Ž 8 đ?‘Š , đ?‘Ž8 = , , , đ?œ•đ?‘Ž8 đ?œ•đ?‘Ž8 đ?œ•đ?‘Ž8 đ?œ•đ?‘Ž8 đ?œ•đ?‘Ž8 = 0 , 0 , 0, −đ?‘¤4 đ?‘‡ . So, đ?‘“đ?‘Ž 8 đ??¸2 , đ?‘Ž8∗ = 0 , 0 , 0 , 0 đ?‘‡ and đ?‘‡

−2 đ?‘Ž4 1 − đ?‘š ≠0 . Thus, according to Sotomayor’s theorem (1.15) system 2 has transcrirtical bifurcation atđ??¸1 with the parameterđ?‘Ž3 = đ?‘Ž3∗

www.ijmret.org

saddle-node bifurcation, nor pitchfork bifurcation can occur at đ??¸2 = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 ,0 .

hence đ?›š 2 đ?‘“đ?‘Ž 8 đ??¸2 , đ?‘Ž8∗ = 0 . Therefore, according to Sotomayor’s theorem (1.15) the saddle-node bifurcation cannot occur. While the first condition of transcritical bifurcation is satisfied. Now, since 0 0 0 0 đ??ˇđ?‘“đ?‘Ž 8∗ đ?‘‹ , đ?‘Ž8 = 0 0 0 0 , 0 0 0 0 0 0 0 −1 where đ??ˇđ?‘“đ?‘Ž 8 đ?‘¤ , đ?‘Ž8 represents the derivative of đ?‘“đ?‘Ž 8 đ?‘‹ , đ?‘Ž8 with respect to đ?‘¤ = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 , đ?‘¤4 đ?‘‡ . Further, it is observed that

Page 7

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016.

đ??ˇđ?‘“đ?‘Ž 8 đ??¸2 , đ?‘Ž8∗ đ?‘‰ 2

0 = 0 0 0

0 0 0 0

0 0 = 0 2 −đ?‘Ł4

�

1 − 2 đ?‘¤2⋇ 3 đ?‘‘12 + đ?‘‘22 3 đ?‘Ł2 , đ?‘Ł33 = đ?‘Ł , đ?‘Ž1 1 − đ?‘š đ?‘¤2⋇ 2 ⋇ − đ?‘‘12 + đ?‘‘22 đ?‘Ž6 đ?‘Ž7 đ?‘¤4 đ?‘Ł43 = đ?‘Ł3, đ?‘‘23 (đ?‘Ž6 + đ?‘¤3⋇ )(đ?‘Ž6 đ?‘Ž8 + (đ?‘Ž8 − đ?‘Ž7 )đ?‘¤3⋇ ) 2 andđ?‘Ł23 any nonzero real number. Heređ?‘Ł13 , đ?‘Ł43

đ?‘Ł12

0 0 0 0 0 0 0 −1

đ?‘Ł13 =

2

đ?‘Ł2

đ?‘Ł32 2

đ?‘Ł4

not equal zero under the existence condition 2.5 đ?‘– .

,

�

Let đ?›š 3 = đ?œ“13 , đ?œ“23 , đ?œ“33 , đ?œ“43 be the eigenvector associated with the eigenvalueđ?œ†4 = 0 of the matrixđ??˝3∗đ?‘‡ .Then we have, đ??˝3∗đ?‘‡ − đ?œ†4 đ??ź đ?›š 3 = 0. By solving this equation forđ?›š 3 we obtain:

đ??ˇđ?‘“đ?‘Ž 8 đ??¸2 , đ?‘Ž8∗ đ?‘‰ 2 = −đ?‘Ł42 đ?œ“42 ≠0 . Now, by substitutingđ?‘‰ 2 in (3.1) we get: đ??ˇ2 đ?‘“ đ??¸2 , đ?‘Ž8∗ đ?‘‰ 2 , đ?‘‰ 2 đ?›š2

−2 đ?‘Ł22

�3

2

đ?œ“23

−2 1 − đ?‘š đ?‘Ł22 đ?‘Ł32 =

2

2đ?‘Ł3

�6 �7 �42 �6 + �3

2

đ?‘Ž4 1 − đ?‘š đ?‘Ł2 + 2

. 2

2

đ?‘Ž6 đ?‘Ž7 đ?‘Ł3 đ?‘Ł4 đ?‘Ž6 + đ?‘¤3 2 Hence, it is obtain that: đ?‘‡ đ?›š 2 đ??ˇ2 đ?‘“ đ??¸2 , đ?‘Ž8∗ đ?‘‰ 2 , đ?‘‰ 2 2

Theorem (3.5):Suppose that the following conditions đ?‘Ž7 > đ?‘Ž8 (1 + 1 − đ?‘š đ?‘Ž6 ) (3.4) are satisfied. Then for the parameter value đ?‘Ž6 đ?‘Ž8 đ?‘Ž2⋇ = 1 − (1 − đ?‘š) đ?‘Ž7 − đ?‘Ž8 system 2.2 at the equilibrium point đ??¸3 = đ?‘¤1∗ , đ?‘¤2∗ , đ?‘¤3∗ , đ?‘¤4∗ has saddle-node bifurcation, but transcrirtical bifurcation, nor pitchfork bifurcation can occur at đ??¸3 . Proof: The characteristic equation given by Eq. 2.12 đ?‘? having zero eigenvalue ( say đ?œ†4 = 0 ) if and only if đ??ś4 = 0 and then đ??¸3 becomes a nonhyperbolic equilibrium point. Clearly the Jacobian matrix of system 2.2 at the equilibrium point đ??¸4 with parameter đ?‘Ž2 = đ?‘Ž2⋇ becomes: đ??˝4∗ = đ??˝ đ?‘Ž2 = đ?‘Ž2⋇ = đ?œ€đ?‘–đ?‘— where 4Ă—4 đ?œ€đ?‘–đ?‘— = đ?‘‘đ?‘–đ?‘— đ?‘“đ?‘œđ?‘&#x; đ?‘Žđ?‘™đ?‘™ đ?‘– , đ?‘— = 1, 2, 3, 4exceptđ?œ€đ?‘–đ?‘— which is đ?œ€22 =

−đ?‘Ž2⋇

đ?‘Ž6 đ?‘Ž7 − 1−đ?‘š . đ?‘Ž7 − đ?‘Ž8

Note that, �2⋇ > 0 provided that condition (3.4) holds . �

Let đ?‘‰ 3 = đ?‘Ł13 , đ?‘Ł23 , đ?‘Ł33 , đ?‘Ł43 be the eigenvector corresponding to the eigenvalueđ?œ†4 = 0 . Thus đ??˝4∗ − đ?œ†4 đ??ź đ?‘‰ 3 = 0 , which gives:

www.ijmret.org

,

Now,

đ?‘Ž6 đ?‘Ž7 đ?‘Ł32 đ?‘Ł42 2 =2 đ?œ“ ≠0 . đ?‘Ž6 + đ?‘¤3 2 4 Thus, according to Sotomayor’s theorem (1.15) system 2.2 has transcrirtical bifurcation atđ??¸2 with the parameterđ?‘Ž8 = đ?‘Ž8∗ .

given by:

đ?œ“23 đ?‘‘12 + đ?‘‘22 3 = − đ?œ“ đ?‘Ž4 (1 − đ?‘š)đ?‘¤3⋇ 2 đ?‘‘12 + đ?‘‘22 đ?œ“3 đ?‘Ž4 (1 − đ?‘š)(đ?‘Ž6 đ?‘Ž8 + (đ?‘Ž8 − đ?‘Ž7 )đ?‘¤3⋇ ) 2 where đ?œ“24 any nonzero real number. đ?œ•đ?‘“ đ?œ•đ?‘Ž 2

= �� 2 � , �2 = �

đ?œ•đ?‘“1

,

đ?œ•đ?‘“2

,

đ?œ• đ?‘“3

,

đ?œ•đ?‘“4 đ?‘‡

đ?œ•đ?‘Ž 2 đ?œ•đ?‘Ž 2 đ?œ•đ?‘Ž 2 đ?œ•đ?‘Ž 2

0 , −đ?‘¤2 , 0 , 0 . đ?‘“đ?‘Ž 2 đ??¸3 , đ?‘Ž2⋇ = 0 , −đ?‘¤2⋇ , 0 , 0

�

=

So,

and hence

�

đ?‘“đ?‘Ž 2 đ??¸3 , đ?‘Ž2⋇ = −đ?‘¤2⋇ đ?œ“24 ≠0 . Therefore, according to Sotomayor’s theorem (1.15) the saddle-node bifurcation occure at đ??¸3 under the existence condition ( 2.5 q ). đ?›š3

The Hopf bifurcation analysis of system đ?&#x;? In this section, the possibility of occurrence of a Hopf bifurcation near the postiveequilibrium point of the system 2 is investigated as shown in the following theorems. Theorem (3.8): Suppose that the following conditions are satisfied: −(đ?‘™ +đ?‘‘ đ?‘‘ ) đ?‘šđ?‘Žđ?‘Ľ −đ?‘™6 , 3 đ?‘‘ 23 32 < đ?‘Ž1 < đ?‘šđ?‘–đ?‘› đ??š where 33

F= đ?‘‘33 ,

đ?‘™3 3(1−đ?‘š )đ?‘¤ 3⋇

max

,

−đ?›¤3 đ?‘™2 đ?‘‘ 34 đ?‘‘ 43

đ??ś3 < min

� 5 � 6 � 7 � 3⋇� 4⋇ � 33 (� 6 +� 3⋇)3

,

đ?‘‘ 33 (−đ?‘Ž 1 đ?‘‘ 12 +đ?‘‘ 33 )

đ?‘™3 , đ?‘Ž1 đ?‘™3 ,2đ?‘‘33 đ?‘™4 + 3đ?‘‘11 <

−đ?‘Ž 1 đ?‘‘ 12 2

,

� 33 +�3 � 5 � 6 � 7 � 3⋇� 4⋇ (� 6 +� 3⋇)3

<

( 3.8 ) đ??ś1 đ??ś2 đ??ś12

, ( 3.9 ) 2 đ?‘™2 đ??ś13 − 4 ∆1 > 0 ( 3.10 ) Then at the parameter value đ?‘Ž2 = đ?‘Ž2 , the system 2.2 has a Hopf bifurcation near the pointđ??¸3 . Proof: According to the Hopf bifurcation theorem (1.17), for n=4, the Hopf bifurcation can occur provided thatđ??śđ?‘– đ?‘Ž2 > 0 ; đ?‘– = 1 ,3 , ∆1 > 0 ,

Page 8

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. đ??ś13 − 4 ∆1 > 0 and ∆2 đ?‘Ž2 = 0 . Straight forward computation gives that: if the following conditions 2.10 đ?‘‘ , 2.10 đ?‘’ ,(3.7),(3.8),(3.9) and (3.10) are hold, thenđ??śđ?‘– đ?‘Ž2 > 0 ; đ?‘– = 1,3 , ∆1 > 0 đ?‘Žđ?‘›đ?‘‘ đ??ś13 − 4 ∆1 > 0 . Now, to verify the necessary and sufficient conditions for a Hopf bifurcation to occur we need to find a parameter satisfy∆2 = 0 . Therefore, it is observed that∆2 = 0 gives that: đ??ś3 đ??ś1 đ??ś2 − đ??ś3 − đ??ś12 đ??ś4 = 0 Now, by using Descartes rule Eq.(3.11) has a unique positive root sayđ?‘Ž2 = đ?‘Ž2 such that: â„’1 đ?‘Ž23 + â„’ 2 đ?‘Ž22 + â„’3 đ?‘Ž2 + â„’4 = 0 â„’1 = đ?‘™1 đ?‘™2 < 0 â„’2 = 1 − đ?‘š đ?‘¤3⋇ đ?‘™1 − đ?‘‘33 đ?‘™2 − đ?‘‘34 đ?‘‘43 đ?‘‘34 đ?‘‘43 + 3đ?‘‘11 + 2đ?‘™1 đ?‘‘33 2 +đ?›¤3 đ?‘™2 + đ?‘‘11 đ?‘‘11 đ?‘™3 − đ?‘‘12 đ?‘‘21 đ?‘‘33 − đ?‘‘33 đ?‘™1 − đ?‘‘12 đ?‘‘21 đ?‘‘33 đ?‘™1 . 2 â„’3 = − 1 − đ?‘š 2 đ?‘¤3⋇ đ?‘™1 đ?‘™2 + 1 − đ?‘š đ?‘¤3⋇ đ?‘‘34 đ?‘‘43 đ?‘‘11 3 + đ?‘™1 2 +2đ?‘‘34 đ?‘‘43 đ?‘‘34 đ?‘‘43 − đ?‘‘33 đ?‘™2 − 2đ?›¤3 đ?‘™2 − 2đ?‘‘11 đ?‘™3 + đ?‘‘12 đ?‘‘21 đ?‘‘33 đ?‘™1 −đ?›¤3 đ?‘‘11 đ?‘™3 + đ?‘™2 + đ?‘‘12 đ?‘‘21 đ?‘‘33 đ?‘™5 + 3 1 − đ?‘š đ?‘¤3⋇ 2 +đ?‘‘33 đ?‘™4 (2đ?‘‘11 − đ?›¤3 + 2 1 − đ?‘š đ?‘¤3⋇ đ?‘‘11 ) â„’4 = 1 − đ?‘š đ?‘¤3⋇ đ?›¤3 đ?‘™1 đ?‘™2 + đ?‘‘12 đ?‘‘21 − 1 − đ?‘š 3 đ?‘¤3⋇ 3 đ?‘™1 đ?‘™2 2 2 +đ?›¤3 đ?‘‘12 đ?‘‘21 đ?‘‘11 + đ?‘‘33 + đ?‘‘33 đ?‘‘34 đ?‘‘43 đ?‘‘12 đ?‘‘21 − đ?‘‘11 ⋇ 2 + 1 − đ?‘š đ?‘¤3 đ?‘‘33 đ?‘™2 + đ?‘™4 − đ?‘™12 đ?‘‘11 − 1 − đ?‘š 2 đ?‘¤3⋇ 2 đ?‘‘11 đ?‘™5 − đ?‘‘33 đ?‘‘12 đ?‘‘21 − đ?‘‘34 đ?‘‘43 − đ?‘‘34 đ?‘‘43 đ?‘‘34 đ?‘‘43 + đ?‘‘11 đ?‘™1 − đ?›¤3 đ?‘‘11 đ?‘‘34 đ?‘‘43 đ?‘™1 − đ?‘‘11 đ?‘™3 2 + đ?‘‘11 + đ?‘‘11 đ?›¤3 đ?‘‘23 đ?‘‘32 − đ?‘‘11 đ?‘‘33 đ?›¤3 đ?‘™6 − đ?‘‘33 đ?‘‘12 đ?‘‘21 đ?‘™1 − đ?›¤3 đ?‘‘11 + 1 − đ?‘š đ?‘¤3⋇ đ?‘‘11 đ?‘‘34 đ?‘‘43 đ?‘™3 + đ?‘‘34 đ?‘‘43 . andđ?‘™1 = đ?‘‘11 + đ?‘‘33 > 0, đ?‘™2 = đ?‘‘34 đ?‘‘43 − đ?‘‘11 đ?‘‘33 > 0 2 đ?‘™3 = đ?‘‘33 − đ?‘‘23 đ?‘‘32 > 0, đ?‘™4 = đ?‘‘34 đ?‘‘43 + đ?‘‘12 đ?‘‘21 <0 2 đ?‘™5 = đ?‘‘33 − đ?‘‘12 đ?‘‘21 > 0, đ?‘™6 = đ?‘‘12 đ?‘‘21 + đ?‘‘23 đ?‘‘32 <0 Now, atđ?‘Ž2 = đ?‘Ž2 the characteristic equation can be

đ?œ†4 1 = đ?›ź1 + đ?‘–đ?›ź2 , đ?œ†4 2 = đ?›ź1 − đ?‘–đ?›ź2 , đ?œ†4 3,4 =

đ??ś3 đ??ś1

đ?œ†24 + đ??ś1 đ?œ†4 +

which has four roots đ?œ†41,2 = Âąđ?‘–

đ??ś3 đ??ś1

∆1 =0 , đ??ś1

� �2 � �2 + � �2 � �2 ≠0 ,

where� , � , �and�are given in (1.29). Note that for �2 = �2 we have �1 = 0 and �2 =

1

∆

1

Clearly, atđ?‘Ž2 = đ?‘Ž2 there are two pure imaginary eigenvalues ( đ?œ†4 1 and đ?œ†4 2 ) and two eigenvalues which are real and negative. Now for all values ofđ?‘Ž2 in the neighborhood ofđ?‘Ž2 ,the roots in general of the following form:

www.ijmret.org

đ??ś3 đ??ś1

,

substituting into (1.29) gives the following simplifications: đ?›š đ?‘Ž2 = −2 đ??ś3 đ?‘Ž2 , đ?›ź2 đ?‘Ž2 đ?›ˇ đ?‘Ž2 = 2 đ??ś1 đ??ś2 − 2 đ??ś3 , đ??ś1 đ??ś3 đ?›Š đ?‘Ž2 = đ??ś4′ đ?‘Ž2 − đ??ś2′ đ?‘Ž2 , đ??ś1 đ??ś3 đ?›¤ đ?‘Ž2 = đ?›ź2 đ?‘Ž2 đ??ś3′ đ?‘Ž2 − đ??ś1′ đ?‘Ž2 đ??ś1

,

where đ??ś1′ =

đ?‘‘đ??ś1 ⃒ =1 , đ?‘‘đ?‘Ž2 đ?‘Ž 2 =đ?‘Ž 2

đ?‘‘đ??ś2 ⃒ = − đ?‘“1 , đ?‘‘đ?‘Ž2 đ?‘Ž 2 =đ?‘Ž 2 đ?‘‘đ??ś3 đ??ś3′ = ⃒ = −đ?‘“2 , đ?‘‘đ?‘Ž2 đ?‘Ž 2 =đ?‘Ž 2 đ?‘‘đ??ś4 đ??ś4′ = ⃒ = đ?‘‘11 đ?‘‘34 đ?‘‘43 . đ?‘‘đ?‘Ž2 đ?‘Ž 2 =đ?‘Ž 2 đ??ś2′ =

Then by using Eq.(1.30) we get that: đ?›Š đ?‘Ž2 đ?›š đ?‘Ž2 + đ?›¤ đ?‘Ž2 đ?›ˇ đ?‘Ž2 = −2đ??ś3 đ?‘Ž2

đ?‘‘11 đ?‘‘34 đ?‘‘43 +

2 đ??ś3( đ?‘™2+đ??ś3đ??ś1) .

đ??ś3 đ?‘™2 đ??ś1

−2

đ?›ź2 đ?‘Ž2 2 đ??ś1

đ??ś1 đ??ś2 −

Now, according to conditions (3.8) and (3.9) we have: � �2 � �2 + � �2 � �2 ≠0 .

So, we obtain that the Hopf bifurcation occurs around the equilibrium point đ??¸3 at the parameterđ?‘Ž2 = đ?‘Ž2 . ∎

V.

andđ?œ†4 3,4 = 2 −đ??ś1 Âą đ??ś12 − 4 đ??ś1 .

.

Clearly, đ?‘…đ?‘’ đ?œ†4đ?‘˜ đ?‘Ž2 ⃒đ?‘Ž 2 =đ?‘Ž 2 = đ?›ź1 đ?‘Ž2 = 0 , đ?‘˜ = 1,2 that means the first condition of the necessary and sufficient conditions for Hopf bifurcation is satisfied atđ?‘Ž2 = đ?‘Ž2 .Now, according to verify the transversality condition we must prove that:

written as: đ?œ†24 +

1 ∆1 −đ??ś1 Âą đ??ś12 − 4 2 đ??ś1

Numerical analysis of system đ?&#x;?

In this section, the dynamical behavior of system 2 is studied numerically for different sets of parameters and different sets of initial points. The objectives of this study are: first investigate the effect of varying the value of each parameter on the dynamical behavior of system 2 and second confirm our obtained analytical results. It is observed that, for the following set of hypothetical parameters that satisfies stability conditions of the

Page 9

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. positive equilibrium point, system 2 has a globally asymptotically stable positive equilibrium point as shown in Fig. 1 . đ?‘Ž1 = 0.7 , đ?‘Ž2 = 0.2 , đ?‘Ž3 = 0.1 , đ?‘Ž4 = 0.5 , đ?‘Ž5 = 0.6, đ?‘š = 0.8 ( 14 ) đ?‘Ž6 = 0.4 , đ?‘Ž7 = 0.5 , đ?‘Ž8 = 0.2 , đ?‘š = 0.8 ( 14 ) 0.65 w11 w12 w13 w14

0.6 0.55

Immature prey

0.5 0.45 0.4 0.35 0.3

starting from four different initial points and this is confirming our obtained analytical results. Now, in order to discuss the effect of the parameters values of system 2 on the dynamical behavior of the system, the system is solved numerically for the data given in 14 with varying one parameter at each time. It is observed that for the data given in 14 with 0.1 ≤ � 1 < 1 , the solution of system 2 approaches asymptotically to the positive equilibrium point as shown in Fig. 2 for typical value � 1 = 0.2 . 1.4

0.25 0.2

0

1

2

3

4

5 Time

6

7

8

9

10

1.2

4

x 10

Immature prey Mature prey Mid- predator Top predator

1

0.8 0.7 0.6

Population

w31 w32 w33 w34

0.8

0.6

Mid-predator

0.5

0.4 0.4

0.2 0.3

0

0.2 0.1 0

0

1

2

3

4

5 Time

6

7

8

9

10 4

x 10

0.9 0.85

w21 w22 w23 w24

0.8

Mature prey

0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4

0

1

2

3

4

5 Time

6

7

8

9

10 4

x 10

1 w41 w42 w43 w44

0.9 0.8

Top predator

0.7 0.6

0

1

2

3

4

5 Time

6

7

9

10 4

x 10

Fig. 2 : Time series of the solution of system 2 for the data given by 14 with đ?‘Ž 1 = 0.2 , which approaches to 1.2 ,0.6 ,0.4 ,0.08 in the interior of đ?‘… 4+ . By varying the parameter đ?‘Ž2 and keeping the rest of parameters values as in 14 , it is observed that for 0.1 ≤ đ?‘Ž2 < 0.44 the solution of system 2 approaches asymptotically to a positive equilibrium point đ??¸3 , while for 0.44 ≤ đ?‘Ž2 < 0.6 the solution of system 2 approaches asymptotically to đ??¸2 = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 ,0 in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 đ?‘¤3 − đ?‘?đ?‘™đ?‘Žđ?‘›đ?‘’ as shownin Fig. 3 for typical value đ?‘Ž2 = 0.5 .

0.5

0.7

0.4

Immature prey Mature prey Mid- predator Top predator

0.6

0.3 0.5

0.2

0

1

2

3

4

5 Time

6

7

8

9

Population

0.1 0

8

10 4

x 10

0.4

0.3

0.2

0.1

0.9 0.85

w21 w22 w23 w24

0.8

0

1

2

3

4

5 Time

6

7

8

9

10 4

x 10

2 for the data given by 14 with đ?‘Ž2 = 0.5 .

0.65 0.6 0.55 0.5 0.45 0.4

0

Fig. 3 :Times series of the solution of system

0.7

0

1

2

3

4

5 Time

6

7

8

9

10 4

x 10

Fig. 1 : Time series of the solution of system 2 that started from four different initialpoints 0.3 , 0.4 , 0.5 , 0.6 , 2.5,0.5 , 2.5 , 1.5 , 2 , 1.5 , 2.5, 2.5 and 2.5 , 2.5, 1.5 , 0.5 for the data given by 2.1 . đ?‘Ž trajectories of đ?‘¤1 as a function of time, đ?‘? trajectories of đ?‘¤2 as a function of time, đ?‘? trajectories of đ?‘¤3 as a function of time, đ?‘‘ trajectories of đ?‘¤4 as a function of time . Clearly, Fig. 1 shows that system 2 has a globally asymptotically stable as the solution of system 2 approaches asymptotically to the positive equilibrium point đ??¸ 3 = 0.34 , 0.6 , 0.4 , 0.08

www.ijmret.org

which approaches to 0.34 , 0.4 , 0.2 , 0 in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 đ?‘¤3 − đ?‘?đ?‘™đ?‘Žđ?‘›đ?‘’ . while for 0.6 ≤ đ?‘Ž2 < 1 the solution of system 2 approaches asymptotically to đ??¸1 = đ?‘¤1 , đ?‘¤2 ,0 ,0 in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 − đ?‘?đ?‘™đ?‘Žđ?‘›đ?‘’ as shown in Fig. 4 for typical value đ?‘Ž2 = 0.8 . 0.7

Immature prey Mature prey Mid- predator Top predator

0.6

0.5

Population

Mature prey

0.75

0.4

0.3

0.2

0.1

0

0

1

2

3

4

5 Time

6

7

8

9

10 4

x 10

Fig. 4 :Times series of the solution of system 2 for the data given by 14 with đ?‘Ž 2 = 0.8 which

Page 10

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016.

approaches

asymptotically to in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 đ?‘¤3 − đ?‘?đ?‘™đ?‘Žđ?‘›đ?‘’. for 0.2 ≤ đ?‘Ž3 < 1 the solution of system ( 2 ) approaches asymptotically to đ??¸1 = đ?‘¤1 , đ?‘¤2 ,0 ,0 in the interior of the positive quadrant of đ??¸2 = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 ,0

đ?‘¤1 đ?‘¤2 − đ?‘?đ?‘™đ?‘Žđ?‘›đ?‘’.

Moreover, varying the parameter đ?‘Ž4 and keeping the rest of parameters values as in 14 , it is observed that for 0.1 ≤ đ?‘Ž4 < 0.26 the solution of system 2 approaches asymptotically to đ??¸1 = đ?‘¤1 , đ?‘¤2 ,0 ,0 in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 − đ?‘?đ?‘™đ?‘Žđ?‘›đ?‘’ ,while for 0.26 < đ?‘Ž4 ≤ 0.4 the solution of system 2 approaches asymptotically to đ??¸2 = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 ,0 in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 đ?‘¤3 − đ?‘?đ?‘™đ?‘Žđ?‘›.and for the 0.4 < đ?‘Ž4 < 1 the solution of system 2 approaches asymptotically to the positive equilibrium point đ??¸3 . For the parameter 0.5 < đ?‘Ž5 ≤ 1 the solution of system 2 approaches asymptotically to a positive equilibrium point đ??¸3 . For the parameter 0.1 ≤ đ?‘Ž6 < 0.2 the solution of system 2 approaches asymptotically to đ??¸2 = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 ,0 in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 đ?‘¤3 − đ?‘?đ?‘™đ?‘Žđ?‘›. While for the 0.2 ≤ đ?‘Ž6 < 0.5 the solution of system 2 approaches asymptotically to a positive equilibrium point đ??¸3 . For the parameters values given in 14 with 0.4 < đ?‘Ž7 < 0.6 the solution of system 2 approaches asymptotically to the positive equilibrium point đ??¸3 . For the parameter 0.1 ≤ đ?‘Ž8 < 0.32 the solution of system 2 approaches asymptotically to the positive equilibrium point đ??¸3 . while for the 0.32 ≤ đ?‘Ž8 < 1 the solution of system 2 approaches asymptotically to đ??¸2 = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 ,0 in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 đ?‘¤3 − đ?‘?đ?‘™đ?‘Žđ?‘›. Moreover, varying the parameter đ?‘š and keeping the rest of parameters values as in 14 , it is observed that for 0.1 ≤ đ?‘š < 0.7 the solution of system 2 approaches asymptotically to the positive equilibrium point đ??¸3 . while for 0.7 ≤ đ?‘š < 0.75 the solution of system 2 approaches asymptotically to đ??¸2 = đ?‘¤1 , đ?‘¤2 , đ?‘¤3 ,0 in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 đ?‘¤3 − đ?‘?đ?‘™đ?‘Žđ?‘›đ?‘’. and for0.75 ≤ đ?‘š < 1

www.ijmret.org

the solution of system ( 2 ) approaches asymptotically to đ??¸1 = đ?‘¤1 , đ?‘¤2 ,0 ,0 in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 − đ?‘?đ?‘™đ?‘Žđ?‘›đ?‘’.

Finally, the dynamical behavior at the vanishing equilibrium point đ??¸0 = 0 ,0 ,0 ,0 is investigated by choosing đ?‘Ž2 = 2 and keeping other parameters fixed as given in 14 , and then the solution of system 2 is drawn in Fig. 5 . 0.7

Immature prey Mature prey Mid- predator Top predator

0.6

0.5

Population

approaches to 0.23 , 0.2 , 0 , 0 in the interior of the positive quadrant of đ?‘¤1 đ?‘¤2 − đ?‘?đ?‘™đ?‘Žđ?‘›đ?‘’ . On the other hand varying the parameter đ?‘Ž3 and keeping the rest of parameters values as in 14 , it is observed that for 0.1 ≤ đ?‘Ž3 < 0.145 the solution of system 2 approaches asymptotically to the positive equilibrium point đ??¸3 . while for0.145 ≤ đ?‘Ž3 < 0.2 the solution of system 2

0.4

0.3

0.2

0.1

0

0

1

2

3

4

5 Time

6

7

8

9

10 4

x 10

Fig. 5 : Times series of the solution of system 2 for the data given by 14 of đ?‘¤đ?‘–đ?‘Ą đ?‘• đ?‘Ž 2 = 2 , which approaches ( 0 , 0 , 0 , 0 ). VI. Conclusions and discussion: In this chapter, we proposed and analyzed an ecological model that described the dynamical behavior of the food chain real system. The model included four non-linear autonomous differential equations that describe the dynamics of four different population, namely first immature prey (đ?‘ 1 ), mature prey (đ?‘ 2 ), mid-predator (đ?‘ 3 ) and (đ?‘ 4 ) which is represent the top predator. The boundedness of system (2.2) has been discussed. The existence condition of all possible equilibrium points are obtain. The local as well as global stability analyses of these points are carried out. Finally, numerical simulation is used to specific the control set of parameters that affect the dynamics of the system and confirm our obtained analytical results. Therefore system (2.2) has been solved numerically for different sets of initial points and different sets of parameters starting with the hypothertical set of data given by Eq.( 4.1 ) and the following observations are obtained. đ?&#x;?. System (2 ) has only one type of attractor in Int. đ?‘…+4 approaches to globally stable point. 2. For the set hypothetical parameters value given in (14 ), the system (2) approaches asymptotically to globally stable positive point đ??¸3 = 0.34 , 0.6 , 0.4 , 0.08 .Further, with varying one parameter each time, it is observed that varying the parameter values, đ?‘Žđ?‘– , đ?‘– = 1,5 đ?‘Žđ?‘›đ?‘‘ 7 do not have any effect on the dynamical behavior of system (2) and the solution of the system still approaches to positive equilibrium pointđ??¸3 = đ?‘¤1∗ , đ?‘¤2∗ , đ?‘¤3∗ , đ?‘¤4∗ . 3. As the natural death rate of mature prey đ?‘Ž2 increasing to 0.43 keeping the rest of parameters as in eq.(14),the solution of system (2) approaches to positive equilibrium point đ??¸3 . However if 0.44 ≤ đ?‘Ž2 < 0.6 , then the top

Page 11

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. predator will face extinction then the trajectory transferred from positive equilibrium point to the equilibrium point E2 = w1 , w2 , w3 ,0 moreover, increasing đ?‘Ž2 ≼ 0.6 will causes extinction in the mid-predator and top predator then the trajectory transferred from equilibrium point E2 = w1 , w2 , w3 ,0 to the equilibrium point E1 = đ?‘¤1 , đ?‘¤2 ,0 ,0 , thus, theđ?‘Ž2 parameter is bifurcation point. 3.As the natural death rate of top predator đ?‘Ž3 increasing to 0.144 keeping the rest of parameters as in eq.(14),the solution of system (2) approaches to positive equilibrium pointđ??¸3 . However if 0.145 ≤ đ?‘Ž3 < 0.2, then the top predator will face extinction then the trajectory transferred from positive equilibrium point to the equilibrium point E2 = w1 , w2 , w3 ,0 moreover, increasing 0.2 ≤ đ?‘Ž3 will causes extinction in the mid-predator and top predator then the trajectory transferred from equilibrium point E2 = w1 , w2 , w3 ,0 to the equilibrium point E1 = đ?‘¤1 , đ?‘¤2 ,0 ,0 , thus, theđ?‘Ž3 parameter is bifurcation point. 4.As the natural death rate of top predator đ?‘Ž4 increasing to 0.1 keeping the rest of parameters as in eq.(14),the solution of system (2) approaches to đ??¸1 = đ?‘¤1 , đ?‘¤2 ,0 ,0 . However if 0.26 ≤ đ?‘Ž4 ≤ 0.4, then the trajectory transferred from đ??¸1 = đ?‘¤1 , đ?‘¤2 ,0 ,0 to the equilibrium point E2 = w1 , w2 , w3 ,0 moreover, increasing đ?‘Ž4 > 0.4 will then the trajectory transferred from equilibrium point E2 = w1 , w2 , w3 ,0 to the positive equilibrium point đ??¸3 = đ?‘¤1∗ , đ?‘¤2∗ , đ?‘¤3∗ , đ?‘¤4∗ , thus, the đ?‘Ž4 parameter is bifurcation point. 5. As the half saturation rate of top predator đ?‘Ž6 decreases under specific value keeping the rest parameters as in eq( 14 ) the top predator will face extinction and the solution of system (2) approaches to E2 = w1 , w2 , w3 ,0 . but where đ?‘Ž6 increases above specific value the trajectory transferred from E2 = w1 , w2 , w3 ,0 to the positive equilibrium point đ??¸3 = đ?‘¤1∗ , đ?‘¤2∗ , đ?‘¤3∗ , đ?‘¤4∗ ,Thus, the parameter đ?‘Ž6 is a bifurcation point. 6. As the natural death rateof the top predatorđ?‘Ž8 ≼ 0.1 increasing keeping the rest of parameters as in eq( 14 ) system (2) has asymptotically stable positive point in Int.đ?‘…+4 .However increasing đ?‘Ž8 ≼ 0.32 will causes extinction in the top predator then the trajectory transferred from positive equilibrium point to the E2 = w1 , w2 , w3 ,0 . Thus, the parameter đ?‘Ž8 is a bifurcation point. 7. As the number of prey inside the refuge m increasing to 0.68 keeping the rest of parameters as in eq.(14), the solution of system (2) approaches to positive equilibrium point đ??¸3 . However if 0.7 ≤ m < 0.75 then the

www.ijmret.org

top predator will face extinction then the trajectory transferred from positive equilibrium point to the equilibrium point E2 = w1 , w2 , w3 ,0 moreover, increasing 0.75 ≤ � will causes extinction in the mid-predator and top predator then the trajectory transferred from equilibrium point E2 = w1 , w2 , w3 ,0 to the equilibrium point E1 = �1 , �2 ,0 ,0 , thus, the� parameter is bifurcation point.

Refrences [1.]

[2.]

[3.]

[4.] [5.]

R.M-Alicea, Introduction to Bifurcation and The Hopf Bifurcation Theorem forplanar System, Colorado State Univesity, 2011. Monk N., Oscillatory expression of hesl, p53, and NF-kappa B driven by transcriptional time delays, Curr Biol.,13:1409-13,2003. L. Perko, Differential equation and dynamical system (Third Editin), New York, Springer-VerlagInc, 2001. Murray J.D. ,“Mathematical biology�, springer-verlag Berlin Heidelberg, 2002. AzharabbasMajeed and Noor Hassan aliThe stability analysis of stage structured prey-predator food chain model with refug,GlobaljournalofEngineringscienceand reserches,ISSN journal of Mothematics, , No. 3 4 ,PP153-164,2016.

Page 12

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.