Numerical study of biological problems in a predator-prey system

The logistic Lotka-Volterra predator-prey equations with diffusion based on Luckinbill's experiment with Didinium nasutum as predator and Paramecium aurelia as prey, have been solved numerically along with a third equation to include prey taxis in the system. The effect of taxis on the dynamics of the population has been examined under three different non-uniform initial conditions and four different response functions of predators. The four response functions are Holling Type 2 Response, Beddington Type Response or Holling Type 3 Response, a response function involving predator interference and a modified sigmoid response function. The operator splitting method and forward difference Euler scheme have been used to solve the differential equations. The stability of the solutions has been established for each model using Routh - Hurwitz conditions, variational matrix. This has been further verified through numerical simulations. The numerical solutions have been obtained both with and without prey-taxis coefficient. The effect of bifurcation value of prey-taxis coefficient on the numerical solution has been examined. It has been observed that as the value of the taxis coefficient is increased significantly from the bifurcation value chaotic dynamics develops for each model. The introduction of diffusion in predator velocity in the system restores it back to normal periodic behaviour. A brief study of coexistence of low population densities both with and without prey-taxis has also been done.