Bifurcation analysis for a size-structured predator-prey model with self-regulation and resource-dependent growth rates

We study a size-structured predator-prey model in which we assume that prey growth rate decreases with its population density, while the predator growth rate depends on predation. We extend the model studied in [Tian, Guo, Iannelli, Pugliese, J. Math. Anal. Appl., 546.2 (2025), p. 129231] by considering size-dependent predation rates: for instance, large prey individuals may be attacked only by large predators, or not at all; small predators may be restricted to small prey, while large predators may avoid small prey individuals due to their low energetic return. Under weak conditions, we prove existence and uniqueness of the solutions of the system. We prove the existence of a predator-free equilibrium, and derive conditions for its stability. We further show that when this equilibrium loses stability, a branch of positive equilibria emerges through a transcritical bifurcation. Finally, we perform numerical bifurcation analysis on the proposed model via pseudospectral reduction. Numerical results show the presence of Hopf bifurcations, both in the branch of prey-only equilibria, and in the branch of coexistence equilibria. Numerical bifurcations show that, depending on the assumptions on predator growth rate, prey population density at the coexistence equilibrium may remain constant, increase or decrease as the prey environment improves, relaxing one part of Rosenzweig's paradox of enrichment.