Multiple dynamics in a delayed predator‐prey model with asymmetric functional and numerical responses

Abstract

We consider a predator‐prey model with dissimilar functional and numerical responses that induce an Allee effect. There is a time lag between consumption and digestion of prey biomass by predator. Hence, a time delay has been incorporated in the numerical response function. The system consists of two interior equilibria. Taking time delay as the bifurcation parameter, four different dynamic behaviors appear, viz., (R1) system undergoes no change in its stability for all time delay, (R2) system undergoes stability change, (R3) system undergoes stability switching, and (R4) system undergoes instability switching. Here, finding four distinct dynamics in a single population model with only one delay is a novelty in this contribution. This variation in dynamics emerges due to asymmetricity in functional and numerical responses. All the relevant theorems in establishing stability are provided, and these are verified numerically. We analytically prove that if an interior equilibrium is a saddle point in absence of time delay, then the equilibrium cannot be stabilized by varying the time delay. It is popularly believed that existence of two distinct pair of purely imaginary roots of the characteristic function leads to stability switching. However, we provide examples where the system remains unstable, stability changes, and instability switching occurs. This is another new and interesting observation in our work. The numerical examples are furnished with phase portraits, time series plots, bifurcation diagrams, and eigenvalues evaluation with delay, for better understanding. Our model with a single delay exhibits variety of dynamics, which were not explored before.

Stability scenarios and period‐doubling onset of chaos in a population model with delayed harvesting

“…In this section, we shall employ the above theory and methodology to determine the critical stability boundaries and the stability regions for stability of the system (2) due to the delayed harvesting. Time delay is intrinsic in many predator-prey systems, and it generally appears in the prey density dependent term or in the functional (or numerical) response term [20][21][22]. Consequently, we cannot vary the time delay parameter arbitrarily in those systems; that is, corresponding to a specific model, we always have a fixed time delay.…”