**Lotka
Volterra Model **

**A Model to simulate Predator - Prey
Interactions **

**Page Designed by : AMIT KUMAR ( Date :
Oct 04 , 2006)**

**Prey = Rabbits**
** Predator = Wolves**

Many of the most interesting dynamics in nature have to do with interactions between organisms. These interactions are often subtle , indirect and difficult to detect. Predator - prey interactions are the exceptions ; predation is often direct, conspicuous and easy to study. For this reason our knowledge of predator - prey relationships is extensive and a large number of models exist to study such interactions . One of such models that simulates predator - prey interactions is the Lotka Volterra Model.

The model is based on simple
assumptions for predator - prey interactions. The first assumption of this model
is that the prey population ** x** , without predation, exhibits simple
exponential growth .

**
**

where **a** is the average birth rate minus the average
death rate of the prey population, and * dx / dt *is the rate
of change of the prey population.

Since predation reduces the prey population, this model
accounts for predation by subtracting some term from equation 1. This term must
account for how often the prey and predator encounter each other. Also, how
often the predator is successful should be accounted for by the model . This
idea is implemented by introducing the simple term **b****xy**
where ** b** is te parameter that indicates the efficiency of predation
and

**
**

In order to define a a related equation for the predator
population, it is assumed that the loss **b****xy **from
the prey population results in an increase in the predator population .This term
when multiplied by a conversion efficiency ( from prey to predator ) shall
account for the growth of the predator population .Thus, the growth rate for the
predator population is

where ** p** is the equal to the product of
conversion efficiency and

** **Since the predators do not live forever, their
death is accounted for by the assumption that the greater is the number of
predators, the greater is the predator death rate ( increased competition
between predators ). Incorporating this idea to equation (3) gives,

**
**

Thus, Equations (2) and (4) form a model for the predator - prey
interactions consisting of two coupled equations. The dynamics of the two
populations depends on the parameters * a , b , p* and

** a** = Intrinsic rate of Prey Population
increase

**Numerical Implementation of the model using Euler's Simplest
Differential Equation Procedure**

To Start with , the above discussed model gives two coupled
differential equations which can easily be solved using the Euler's technique of
solving differential equations . The formula used for this technique was

**
**which gives the solution for the equations numerically. The
Parameters chosen for this simulation was a = 1 , b = 0.2 , p = 0.04 and c =
0.5. The initial prey population was chosen to be 5 and the Predator population
was chosen to be 2 . The time step was chosen as 0.01. Clearly, the solution
derived from this model

Java Code : LotkaEulers.java
StdDraw.java

Please Download Both these files to a folder and run the LotkaEulers.java
file . It won't run till StdDraw.java is in the same folder as StdDraw.java is a
dependancy for LotkaEulers.java .The StdDraw.java file helps in auto-genearting
plots on the Screen

**Numerical Implementation of the model using Runge Kutta
Differential Equation Solver Procedure**

The coupled differential equations in the Lotka Volterra Model were implemented using the Fourth order Runge-Kutta numerical technique for solving differential equations.

The fourth order Runge-Kutta requires four gradient or ''k'' terms to calculate .

The Parameters chosen for this simulation were a = 1 , b = 0.2 , p = 0.04 and c
= 0.5. The initial prey population was chosen to be 5 and the Predator
population was chosen to be 2 . The time step was chosen as 0.01. As can be seen
, the phase space obtained from the output of the Runge-Kutta program does
not seem to spread with time, which is an indication that this technique is
better suited for the implementation of the model than Euler's technique.The
predator and the prey Population cycle with time. Also, the predator population
follows the prey population. Though this cycling behavior predicted by this
model occurs in nature , it is not very common. Thus, other variants of this
model have been used to simulate predator -Prey interactions.

Java Code : LotkaKutta.java
StdDraw.java

Please Download Both these files to a folder and run the LotkaKutta.java
file . It won't run till StdDraw.java is in the same folder as StdDraw.java is a
dependancy for LotkaKutta.java .The StdDraw.java file helps in auto-genearting
plots on the Screen

**Numerical Implementation of Competitive Lotka Volterra model **

As both the predator and the prey compete for food and shelter in the forest, competition sets in and the population of each species tends to control itself via a negative quadratic effect , that is the population decreases with a rate directly proportional to the present population of that species. Hence , higher the population , higher is the quadratic decrease. This effect can be introduced in the model by incorporating a quadraic term in each of equations (2) and (4) as follows :

This effect was introduced in the model and the coupled differential
equations were again solved using Fourth Order Runge- Kutta Differential
equation Solver Procedure.The Parameters chosen for this simulation were a = 1 ,
b = 0.2 , p = 0.04 and c = 0.5. The initial prey population was chosen to be 5
and the Predator population was chosen to be 2 . The time step was chosen as
0.01 .The value of a was chosen as 0.01 and
that of b was chosen to be 0.005. As can
be seen , the phase space obtained from the output of the Runge-Kutta
program converges with time towards a solution where the prey and the predator
Population becomes constant with time. This is the predicted behaviour to be
expected when competition sets in Predator-Prey interactions.

Java Code :
LotkaVar.java
StdDraw.java

Please Download Both these files to a folder and run the LotkaVar.java file
. It won't run till StdDraw.java is in the same folder as StdDraw.java is a
dependancy for LotkaVar.java .The StdDraw.java file helps in auto-genearting
plots on the Screen

**Numerical Implementation of Competitive Lotka Volterra model with
Recurring Hunters**

Let us assume that the forest in which the Rabbits and Wolves live is pretty close to a village . The villagers are wary of the wolves attacking their cattle in the night and also do not want to kill them completely ( they want to avoid huge rise in rabbit population which can affect their crops ) . Thus, they decide to send hunters who kill the Wolves if their number exceeds a certain critical value (y0). This idea can be incorporated in the model by applying the following condition .

This effect was introduced in the model and the coupled differential
equations were again solved using Fourth Order Runge- Kutta Differential
equation Solver Procedure. The Parameters chosen for this simulation were a = 1 ,
b = 0.2 , p = 0.04 and c = 0.5. The initial prey population was chosen to be 5
and the Predator population was chosen to be 2 . The time step was chosen as
0.01 .The value of b was chosen to be 0.1
and the critical number of the wolves ( Predators ) was set at y0 = 3. As can
be seen , the phase space obtained from the output of the Runge-Kutta
program converges very quickly with time in comparison to the previously defined
cases . Both the prey and the predator
Population seem to become constant with time. This is the predicted behaviour to be
expected in Predator-Prey interactions when external conditions are
applied on the population of a certain species.

Java Code :
LotkaHunters.java
StdDraw.java

Please Download Both these files to a folder and run the LotkaHunters.java file
. It won't run till StdDraw.java is in the same folder as StdDraw.java is a
dependancy for LotkaVar.java .The StdDraw.java file helps in auto-genearting
plots on the Screen

Thus, this assignment helps us in understanding the practical application
of numerical solution to differential solutions . Even very complex situations
can be simulated by applying the correct conditions and using the appropriate
numerical technique for that problem.

References :

www.sumanasinc.com/webcontent/