Article Description

(1)

Volume 1, Issue 8 (October2012) PP: 26-30

PREY-PREDATOR MODELS IN MARINE LIFE

Prof.V.P. Saxena

1

, Neeta Mazumdar

2

1_{Prof. V.P. Saxena, Director, Sagar Institute of Research, Technology and Science, Bhopal-462041, India.} 2_{Ms.Neeta Mazumdar, Associate professor, S.S. Dempo College of Commerce and Economics, Altinho, Panaji}

Goa

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:



































r

N

rD

r

P

K

b

K

N

r

t

N

_i

i i

i i i i i

i

i 1

1 1

1 1 1

1

1 



































r

P

rD

r

N

K

b

P

r

t

P

_i

i i

i i i

i

i 1

2 2

1

,

i=1, 2 (1)

The total area is divided into two patches. The first patch is assumed to lie along the radius

r

₀



r



r

₁ and second patch lies along the radius

r

₁



r



r

₂ .

th

(2)

i

N

₁ =Density of prey population

i

P

₁ =Density of predator population

i

r

₁ =Intrinsic growth rate of prey population

i

r

₂ =Intrinsic growth rate of predator population

i

b

₁ =Interspecific interaction coefficient

i

b

₂ = Interspecific interaction coefficient

i

D

₁ ,

D

₂_i=Diffusion coefficient of prey and predator populations (i=1,2)

Equilibrium points of the equations (1) are

 

0 ,

0

1i

E

,

E

₂_i



N

_i

,

P

_i



where,

i i i

b

K

N

2 2





,

i i i

i i

b

K

b

K

P

1 2

2 1

1







,i=1,2

We take small perturbations

i

u

₁ and

i

u

₂ from the non zero equilibrium point i.e.

N

_i



N

_i



u

₁_i,

P

_i



P

_i



u

₂_i (2) where

u

₁_i



1

,

u

₂_i



1

and

u

₁_i,

u

₂_i are prey and predator populations. Then system of equations (1) becomes





































r

u

rD

r

K

u

b

K

u

N

r

t

u

_i

i i

i i i i

i 1

1 1

2 1 1 1 1

1

1 

































r

u

rD

r

K

u

b

P

r

t

u

_i

i i

i i i i

i 2

2 2

1 2 2

2

1

(3)

Initial conditions are taken as

u

₁_i

 

r

,

0 

G

₁_i

 

r

,

u

₂_i

 

r

,

0 

G

₂_i

 

r

(4) Where

G

_ji

 

r

are known functions.

Interface conditions at

r



r

₁are assumed to be

2 1

12 12 11

11

R

r

u

D

r

u

D









2 1

22 22 21

21

R

r

u

D

r

u

D









(5)

(3)

2 0

12 11

r r r

r

u

r

u









2 0

22 21

r r r

r

u

r

u









(6)

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

 



A

u

B

u

C

u

D

u

E

u



r

F

_i



_i ₁2_i



_i ₂2_i



_i ₁_i ₂_i



_i ₁'_i 2



_i ₂'_i 2 , i=1,2 (7)

where

r

u

_i i





1

'

1 ,

r

u

_i i





2

'

2 , i=1,2

Here,





















i i i

K

N

r

t

A

1 1 1

2

1 t

B

_i





2

1

i i i i

K

b

P

r

K

b

N

r

C

i

2 2 2 1

1

1 







2

1i i

D





,

2

2i i

D

E





The corresponding variational form is







i i

L

L i

i

F

dr

I

1

(8) We take

r

B

A

u

₁_i



i



i

log

u

₂_i



C

i



D

i

log

r

(9) 0

110

11

u

at

r

L

u



u

₁₁



u

₁₁₁



u

₁₂

at

r



L

₁ (10)

2 112

12

u

at

r

L

u



u

₂₁



u

₂₂₀

at

r



L

₀

u

₂₁



u

₂₂₁



u

₂₂

at

r



L

₁ (11)

2 222

22

u

at

r

L

u



Using these values we get

 

1

1 11

log

  





i i

i

L

u

L

u

A

 



1



1 11 11

log

_





i i

i

L

u

B

 

1

1 22

log

  





i i

i

L

u

L

u

(4)

 

22

log

22

 

1 ₁

log

_ 





i i i i i

L

u

D

(12)

2 ,

1 

i

Accordingly

 





 



i



i i i i i i i i i i i i i i i i

D

C

D

C

B

A

B

A

I

3 2 2 1 2 3 2 2 1 2

2

2 











 

i

i i i i i i i i i i i i i i i i i

D

E

B

D

C

B

D

A

D

B

C

A

C

4 2 4 2 3 2 1

)

(





(13)

where

I



I

₁



I

₂ (14) Now differentiating I with respect to nodal points 111

u

and

u

₂₂₁; and putting

0

111





u

I

,

0

221





u

I

, we get



1 17 1 18



111



1 11 2 12 1 13 2 14



112



2 110 2 111



110

A



D



u

A



A



D



D



u

A



D



u





₁ ₁₉



₂₂₁



₁ ₁₅ ₂ ₁₆



₂₂₂



₂ ₁₁₂



0

220







u

C



u

C



C



u

C



(15)



1 27



111



1 21 2 22



112



2 210



220



1 28 1 29



110

C



u

C



C



u

C



u

B



E



u





₁ ₂₃ ₂ ₂₄ ₁ ₂₅ ₂ ₂₆



₂₂₂



₂ ₂₁₁ ₂ ₂₁₂



0

221







u

B



B



E



E



u

B



E



(16)

Substituting values of

A

_i

,

B

_i

,

C

_i

,

D

_i

,

E

_i,(i=1,2) and taking Laplace transform of (15)and (16), we get



₁₁ ₁₂



₁₁₁ ₁₃ ₂₂₁

1

₁₄ ₁₁

u

₁₁₁

 

0 p

u

p









(17)



₂₂ ₂₃



₂₂₁

1

₂₄ ₂₂ ₂₂₁

 

0

111

21

u

p

u

p

u









(18)

Here

u

₁₁₁, 221

u

are Laplace transforms of

u

₁₁₁, 221

u

respectively and

222 220 112

110

,

u

,

u

,

u

are independent

of t and

u

₁₁₁

 

0

and

u

₂₂₁

 

0

are initial values. Solving (17 and (18), we get,

)

(

)

0 (

)

0 (

(

13 21 23 12 22 12 23 11 2 22 11 13 24 23 14 221 13 22 111 23 11 22 14 2 22 11 111



















p

u

p

u

p

u

)

(

)

0 (

)

0 (

)

0 (

(

13 21 23 12 22 12 23 11 2 22 11 12 24 21 14 221 12 22 111 21 11 11 24 2 221 22 11

221

_













p

u

p

u

p

u

(19) Taking inverse Laplace transform of

111

u

and

221

u

,we obtain the expressions for 111

u

and

221

u

.Substituting these

values in (13), we get values of

u

_{1 and}_i

u

_{2 (i=1,2).Using these values we can get the values of}_i

N

_i and

P

_i(i=1,2).

IV. NUMERICAL COMPUTATION

(5)

04 .

12



r

,

b

₁₁



.

5

,

b

₁₂



.

4

,

K

₁₁



100

,

K

₁₂



125

,

d

₁₁



.

9

,

d

₁₂



.

8

,

r

₂₁



.

025

,

035 .

22



r

,

b

₂₁



.

5

,

b

₂₂



.

4

,

K

₂₁



110

,

K

₂₂



130

,

d

₂₁



.

8

,

d

₂₂



.

9

,

r

₀



10

,

r

₁



20

,

30

2



r

,

u

₁₁₁₀



.

5

,

u

2210



.

5

,

u

1112



.

9

,

u

222



.

9

,

u

110



.

2

,

u

220



.

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.

REFERENCES

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

2. Adhikari, J.N., Sinha, B.H. Fundamentals of biology of animals, New Central Book Agency, Calcutta, India, 1980, pp.117-153.

3. 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.

4. Cushing, J. M. Predator- prey interactions with time-delays. Journal of Mathematical Biology, 1976, 3, pp.369-380.

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

6. Cantrell, R.S. and Cosner, C. Diffusive logistic equations with indefinite weights: Population model in disrupted environments, 1989, pp.101-123.