How to add a partial differential equation to lotka volterra equation

Hello, I am working on the Lotka-Volterra predator-prey model in the attached files.

Description of the predato - pray model in the attached files as follows

 y1' = (1 - alpha*y2)*y1
 y2' = (-1 + beta*y1)*y2

The functions y1 and y2 measure the sizes of the prey and predator populations respectively. The quadratic cross term accounts for the interactions between the species. Note that the prey population increases when there are no predators, but the predator population decreases when there are no prey.

I need to extend the attached model with the diffusion of the both population in the medium through single coordinate (1-D plane) as shown in the below diagram. Hence I need to add the diffusion of the organisms through the 1- D plan to the model. This model can be describe with a partial differential equation adding to the Lotka-Volterra predator-prey system provide diffusion of organisms in the Lotka-Volterra predator-prey system

dμ/(dt) = D1 . (d^2 μ)/(dX^2) + μ . (1-μ/K_1 + beta_1_2 . V/K_1) (1) //μ-population of one organism, D1- diffusion of the organism(constant), X - coordinate, r1 - growth rate(constant)

dV/(dt) = D2 . (d^2 V)/(dX^2) + V . (-V/K_2 + beta_2_1 . μ/K_2) (2)

In the attached files I have implemented the unbold part of the equation (i.e Lotka-Volterra predator-prey model). I need to add the bold part in the equation to the attached model. I would appreciate a lot if you could help me to add the bold parts in the (1) and (2) equation to the attached files. Please assume any values for constant (ex. D1 and D2 = 1).

Thank you in advance!

Answer 1

[Arthur Goldsipe]

Hi, SimBiology does not currently support PDEs. If you want to model this system in SimBiology, you would need to discretize the X coordinate and convert the system into an equivalent set of ODEs.

[zahmeeth sakkaff]

Thank you for directing me to a write direction

Answer 2

[Sreeja Banerjee]

Hi Zahmeeth,

From your question I understand that you want to know how you can write the second order partial differential equations for your model. I wanted to point you to some MATLAB documentation in that regard:

1. You may use the PDEPE function to solve first and second order parabolic-elliptic PDEs. Please look at the documentation at the following links:

a) http://www.mathworks.com/help/matlab/math/partial-differential-equations.html

b) http://www.mathworks.com/help/matlab/ref/pdepe.html

2. There is an example in the MATLAB documentation on Stochastic Simulation of the Lotka-Volterra Reactions. However, you will need SimBiology toolbox for performing this. Please find the documentation at the following link: http://www.mathworks.com/help/simbio/examples/stochastic-simulation-of-the-lotka-volterra-reactions.html