E0182630

International Journal of Engineering Inventions ISSN: 2278-7461, www.ijeijournal.com Volume 1, Issue 8 (October2012) PP: 26-30

PREY-PREDATOR MODELS IN MARINE LIFE Prof.V.P. Saxena 1, Neeta Mazumdar2 1

Prof. V.P. Saxena, Director, Sagar Institute of Research, Technology and Science, Bhopal-462041, India.

Ms.Neeta Mazumdar, Associate professor, S.S. Dempo College of Commerce and Economics, Altinho, Panaji Goa

2

Abstract:––The paper discusses the mathematical model based on the prey-predator system, which diffuses in a near linear bounded region and the whole region is divided into two near annular patches with different physical conditions. The model has been employed to investigate the population densities of fishes depending on time and position. Finite Element Method has been used for the study and computation. The domain is discretized into a finite number of sub domains (elements) and variational functional is derived. Graphs are plotted between the radial distance and the population density of species for different value of time.

Key words:––Discretization, diffusion, intrinsic growth, Laplace transform, variational form.

I.

1 INTRODUCTION

The prey-predator relationship is the most well known of all the nutrient relationships in the ocean. [2] Fishes as a group provide numerous examples of predators. Thus the herbivorous fishes such as the Sardines and the Anchovy are sought after by predators of the pelagic region. These plankton feeding fishes form an important link between the plankton and the higher carnivores which have no means of directly utilizing the rich plankton as a source of food. Many of the predatory fishes, such as tuna, the barracuda and the salmon, have sharp teeth and they move at high speed in pursuit of their prey.[7] Many marine mammals such as the killer whales, porpoises, dolphins, seals, sea lions and walruses are all predators with well developed sharp teeth as an adaptation for predatory life. The pelagic fishes of great depths resort to predatory life as they have other source of food there. But owing to the scarcity of food available, they are comparatively smaller than the predators of the surface region. These fishes have developed many interesting contuvances and food habits. Thus many of the deep sea fishes such as the chiasmodus have disproportionately large mouths and enormously distensible stomachs and body walls which permit swallowing and digestion of fishes up to three times their own size. The mouth is well armed with formidable teeth to prevent the escape of the prey. Some of these fishes are provided with luminescent organs and special devices to make the best of the prevailing conditions. Many benthic ferns also are predaceous, living upon each other and other bottom animals.[5] Bottom fishes, especially the ray, live on crustaceans, shell fish, worms and coelentrates of the sea floor.

II.

MATHEMATICAL MODEL

In this paper, we discuss the mathematical model based on the prey-predator system, which diffuses in a near linear bounded region and the whole region is divided into two patches. The model has been employed to investigate the population densities of fishes depending on time and position.[4] Finite Element Method has been used for the study and computation. This approach can be extended to the study of population in more patches.

Variational finite element method is also employed.[1] This method is an extension of Ritz method and is used for more complex boundary value problems. In the variational finite element method the domain is discretized into a finite number of sub domains (elements) and variational functional is obtained.[3],[6] The approximate solution for each element is expressed in terms of undetermined nodal values of the field variable, as appropriate shape functions (trial functions) or interpolating functions.

Here we have taken a prey-predator model, in which prey-predator populations diffuse between two circular patches in a given area. The system of non-linear partial differential equations for the above case is:



  1  N i N b N   r1i N i 1  i  1i Pi    rD1i 1i  t r   K1i K1i  r r 

  1  Pi b P   r2i Pi   1  2i N i    rD2i 1i  , t K 2i  r r  r   i=1, 2

(1)

The total area is divided into two patches. The first patch is assumed to lie along the radius r0 patch lies along the radius r1 th Here, for i patch we take ISSN: 2278-7461

 r  r2

 r  r1

and second

.

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PREY-PREDATOR MODELS IN MARINE LIFE

N1i =Density of prey population P1i =Density of predator population r1i =Intrinsic growth rate of prey population r2i =Intrinsic growth rate of predator population b1i =Interspecific interaction coefficient b2i = Interspecific interaction coefficient D1i , D2i =Diffusion coefficient of prey and predator populations

(i=1,2)

Equilibrium points of the equations (1) are   E1i 0,0 , 2i i i where,

  E N , P



K 2i  K1i K , Pi   2 ,i=1,2 b2i b1i b2i b1i We take small perturbations u1i and u2i from the non zero equilibrium point i.e.  N i  N i  u1i , Pi  Pi  u2i (2) u1i  1, u2i  1 and u1i , u 2i are prey and predator populations.

N i 

where

Then system of equations (1) becomes

u u1i bu  1  u   r1i N i  1i  1i 2i    rD1i 1i  t K1i  r r  r   K1i b u  1   u2i u   r2i Pi   2i 1i    rD2i 2i  t r   K 2i  r r 

(3)

Initial conditions are taken as

u1i r ,0  G1i r , u2i r ,0  G2i r  G ji r  are known functions.

Where

(4)

r  r1 are assumed to be u u D11 11   D12 12 r R1 r

Interface conditions at

D21

u 21 r

  D22 R1

u 22 r

R2

(5) R2

R1 and R2 denote region –I and region-II respectively.

Boundary conditions associated with the system of equations are where

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PREY-PREDATOR MODELS IN MARINE LIFE

u11 r

 r  r0

u12 r

r  r2

u 21 r

 r  r0

u 22 r

(6) r  r2

III.

SOLUTION

To solve this model, we apply the Finite Element Method. Comparing the system of equations (3) with Euler Lagrange’s equations, we get 2 2 Fi  Ai u12i  Bi u22i  Ci u1i u2i  Di u1' i  Ei u2' i r , i=1,2 (7)



 

 

u1i ' u 2i , u 2i  r r  1  N  Here, Ai    r1i 1  2  t K1i  1  Bi  2 t  r1i N i b1i r2i Pib2i Ci   K1i K 2i u1' i 

where

, i=1,2

Di  

D1i D2i , Ei   2 2

The corresponding variational form is Li

Ii 

 Fi dr

(8)

We

take

Li 1

u1i  Ai  Bi log r

u2i  C i  Di log r u11  u110 at r  L0 u11  u111  u12 at r  L1 u12  u112 at r  L2 u 21  u 220 at r  L0 u 21  u 221  u 22 at r  L1 u 22  u 222 at r  L2

(9)

(10)

(11)

Using these values we get

Ai  Bi 

Ci  ISSN: 2278-7461

u11i 1 log Li   u11i log Li 1  log Li   log Li 1  u11i  u11i 1

log Li   log Li 1  u 22i 1 log Li   u 22i log Li 1 

log Li   log Li 1 

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PREY-PREDATOR MODELS IN MARINE LIFE

Di 

i  1,2



Accordingly

u 22i  u 22i 1

log Li   log Li 1 



 

2

(12)

2

I i  Ai Ai 1i  B i  2i  2 Ai B i 3i







 

2

2

 Bi C i 1i  D i  2i  2C i D i 3i



 Ci A C 1i  B D  2i  ( A D  B C ) 3i i

i

i

i

i

i

 

 Di B

i 2

4i

i

i



 

 Ei D

 (13)

i 2

4i

I  I1  I 2 with respect to nodal points u111 and u 221 ; and putting where

I I  0,  0, u111 u221

(14)

Now differentiating I

we get

u110  A1 17  D1 18   u111  A1 11  A2 12  D113  D2 14   u112  A2 110  D2 111   u 220 C1 19   u 221 C1 15  C 2 16   u 222 C 2 112   0 u110 C1 27   u111 C1 21  C2  22   u112 C2  210   u 220 B1 28  E1 29   u 221 B1  23  B2  24  E1  25  E 2  26   u 222 B2  211  E 2  212   0 (16)

(15)

Ai , Bi , Ci , Di , Ei ,(i=1,2) and taking Laplace transform of (15)and (16), we get  11 p   12 u111   13u221  1  14   11u111 0 (17) p 1  21u111   22 p   23 u221   24   22u221 0 (18) p Substituting values of

u

Here 111 , of t and

u221 are Laplace transforms of u111 , u221 respectively and u110 , u112 , u220 , u222 are independent

u111 0  and u 221 0  are initial values.

u111 

Solving (17 and (18), we get,

( 11 22 p   14 22 p   11 23u111 (0) p   22 13u 221 (0)   14 23   24 13 ) p( 11 22 p 2   11 23 p   12 22 p   12 23   21 13 ) 2

( 11 22 u 221 (0) p 2   24 11 p   11 21u111 (0) p   22 12 u 221 (0) p u 221 

  14 21   24 12 ) p( 11 22 p 2   11 23 p   12 22 p   12 23   21 13 ) (19)

u111 and u221 ,we obtain the expressions for u111 and u221 .Substituting these values in (13), we get values of u1i and u 2i (i=1,2).Using these values we can get the values of Ni and Pi (i=1,2).

Taking inverse Laplace transform of

IV.

NUMERICAL COMPUTATION

We make use of the following values of parameters and constants.

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PREY-PREDATOR MODELS IN MARINE LIFE

r12  .04 , b11  .5 , b12  .4 , K11  100 , K12  125 , d11  .9 , d12  .8 , r21  .025 , r22  .035 , b21  .5 , b22  .4 , K 21  110 , K 22  130 , d 21  .8 , d 22  .9 , r0  10 , r1  20 , r2  30 , u1110  .5 , u2210  .5 , u1112  .9 , u222  .9 , u110  .2 , u220  .2 .

V.

CONCLUSION

Graphs are plotted between radius r and population density of species for different values of time. Graphs show that density of prey increases with radius while density of predator decreases with radius at any time in the first patch. In the second patch density of prey decreases with radius while density of predator increases. Graphs also show that density of prey increases with time and density of predator decreases with time in the first patch. In the second patch density of prey decreases with time while density of predator increases with time.

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Ainseba and Anita, Internal exact controllability of the linear population dynamics with diffusion, Elec. Journal of Differential Equations, 2004, 112, pp.1-11.

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Bhattcharya, Rakhi, Mukhopadhyay, B. and Bandyopadhyay, M., Diffusive instability in a Prey-Predator system with time dependent diffusivity. International Journal of Mathematics And Mathematical Sciences, 2002,40, pp.4195-4203.

Cushing, J. M. Predator- prey interactions with time-delays. Journal of Mathematical Biology, 1976, 3, pp.369380.

Cushing, J. M and Saleem, M., A predator-prey model with age- structure, Journal of Mathematical Biology, 1982,14, pp.231-250.

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