Numerical approach to an age-structured Lotka-Volterra model

We study the impact of an age-dependent interaction in a structured predator-prey model. We present two approaches, the PDE (partial differential equation) and the renewal equation, highlighting the advantages of each one. We develop efficient numerical methods to compute the (un)stability of steady-states and the time-evolution of the interacting populations, in the form of oscillating orbits in the plane of prey birth-rate and predator population size. The asymptotic behavior when species interaction does not depend on age is completely determined through the age-profile and a predator-prey limit system of ODEs (ordinary differential equations). The appearance of a Hopf bifurcation is shown for a biologically meaningful age-dependent interaction, where the system transitions from a stable coexistence equilibrium to a collection of periodic orbits around it, and eventually to a stable limit cycle (isolated periodic orbit). Several explicit analytical solutions are used to test the accuracy of the proposed computational methods ​
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