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)*y2The 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!
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