This document summarizes a mathematical model of prey-predator populations in marine life. The model divides a bounded region into two circular patches with different conditions. Finite element methods are used to study how fish population densities change over time and position. Equations model the diffusion and growth of prey and predator populations between patches. The domain is discretized and a variational formulation is derived. Graphs of population density versus radial distance can be plotted for different times.

1. 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.
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:
N i  N b  1  N 
 r1i N i 1  i  1i Pi  
 K  rD1i 1i 
t  1i K1i  r r 
 r 
Pi  b  1  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  r  r1 and second
patch lies along the radius r1  r  r2 .
th
Here, for i patch we take
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2. PREY-PREDATOR MODELS IN MARINE LIFE
N1i =Density of prey population
P i =Density of predator population
1
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
  E N , P
E1i 0,0 , 2i

i i


where,
K 2i  K1i K
N i  , P i  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 , P  P  u2i
i i (2)
where u1i  1, u2i  1 and u1i , u 2i are prey and predator populations.
Then system of equations (1) becomes
u1i u bu  1  u 
 r1i N i  1i  1i 2i  
K  rD1i 1i 
t  1i K1i  r r 
 r 
u2i b u  1   u 
 r2i Pi   2i 1i  
 K  r r  rD2i 2i  (3)
t  2i   r 
Initial conditions are taken as
u1i r ,0  G1i r ,
u2i r ,0  G2i r  (4)
Where G ji r  are known functions.
r  r1 are assumed to be
Interface conditions at
u u
D11 11   D12 12
r R1 r R2
u 21 u 22
D21   D22 (5)
r R1 r R2
where R1 and R2 denote region –I and region-II respectively.
Boundary conditions associated with the system of equations are
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3. PREY-PREDATOR MODELS IN MARINE LIFE
u11 u12

r r  r0 r r  r2
u 21 u 22
 (6)
r r  r0 r 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
Fi  Ai u1i  Bi u2i  Ci u1i u2i  Di u1i  Ei u2i r , i=1,2
2 2
 
' 2
(7)
u1i ' u 2i
where u1i 
, u 2i 
'
, i=1,2
r r

1  N 
Ai    r1i 1 
2  t K1i 
Here,
 
1 
Bi 
2 t

r1i N i b1i r2i Pib2i D1i D2i
Ci   Di   , Ei  
K1i K 2i 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 (9)
u11  u110 at r  L0
u11  u111  u12 at r  L1 (10)
u12  u112 at r  L2
u 21  u 220 at r  L0
u 21  u 221  u 22 at r  L1 (11)
u 22  u 222 at r  L2
Using these values we get
u11i 1 log Li   u11i log Li 1 
Ai 
log Li   log Li 1 
u11i  u11i 1
Bi 
log Li   log Li 1 
u 22i 1 log Li   u 22i log Li 1 
Ci 
log Li   log Li 1 
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4. PREY-PREDATOR MODELS IN MARINE LIFE
u 22i  u 22i 1
Di 
log Li   log Li 1 
(12)
i  1,2
 
Accordingly

2
 
I i  Ai Ai 1i  B i  2i  2 Ai B i 3i
2
 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 i i

   
(13)
i 2 i 2
 Di B 4i  Ei D 4i
where I  I1  I 2 (14) Now differentiating I
with respect to nodal points u111 and u 221 ; and putting
I I
 0,  0, we get
u111 u221
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 (15)
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)
Ai , Bi , Ci , Di , Ei ,(i=1,2) and taking Laplace transform of (15)and (16), we get
Substituting values of
 11 p   12 u111   13u221  1  14   11u111 0 (17)
p
 21u111   22 p   23 u221   24   22u221 0
1
(18)
p
u
Here 111 , u221 are Laplace transforms of u111 , u221 respectively and u110 , u112 , u220 , u222 are independent
of t and u111 0  and u 221 0  are initial values. Solving (17 and (18), we get,
( 11 22 p   14 22 p   11 23u111 (0) p   22 13u 221 (0)   14 23   24 13 )
2
u111 
p( 11 22 p 2   11 23 p   12 22 p   12 23   21 13 )
( 11 22 u 221 (0) p 2   24 11 p   11 21u111 (0) p   22 12 u 221 (0) p
  14 21   24 12 )
u 221 
p( 11 22 p 2   11 23 p   12 22 p   12 23   21 13 )
(19)
Taking inverse Laplace transform of 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 P (i=1,2).
i
IV. NUMERICAL COMPUTATION
We make use of the following values of parameters and constants.
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5. 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.
REFERENCES
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1980, pp.117-153.
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