Mathematical Analysis of a Prey Predator and Ammensal Model with Time Delay

We study a three-species ecological model in which the first species (x) preys on the second species(y) and the second species exerts an amensal effect on the third species (z). Interactions between predator and prey include an explicit time delay(τ) and the system is formulated as a set of delay differential equations. We identify the positive coexistence equilibrium and perform a local stability analysis about this steady state. we derive sufficient conditions for a Hopf bifurcation driven by the delay parameter (τ). By treating (τ) as the bifurcation parameter, we determine critical delay values at which the coexistence equilibrium loses stability and periodic oscillations emerge. Numerical simulations implemented in MATLAB confirm the analytical predictions and illustrate the instability regimes and bifurcating limit cycles.

Prey, Predator Co-existing state, local stability, Hopf bifurcation

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