A predator-prey model with age-structured role reversal

We propose a predator-prey model with an age-structured predator population that exhibits a functional role reversal. The structure of the predator population in our model embodies the ecological concept of an “ontogenetic niche shift,” in which a species’ functional role changes as it grows. This structure adds complexity to our model but increases its biological relevance. The time evolution of the age-structured predator population is motivated by the Kermack-McKendrick Renewal Equation (KMRE). Unlike KMRE, the predator population’s birth and death rate functions depend on the prey population’s size. We establish the existence, uniqueness, and positivity of the solutions to the proposed model’s initial value problem. The dynamical properties of the proposed model are investigated via Latin Hypercube Sampling in the 15-dimensional space of its parameters. Our Linear Discriminant Analysis suggests that the most influential parameters are the maturation age of the predator and the rate of consumption of juvenile predators by the prey. We carry out a detailed study of the long-term behavior of the proposed model as a function of these two parameters. In addition, we reduce the proposed age-structured model to ordinary and delayed differential equation (ODE and DDE) models. The comparison of the long-term behavior of the ODE, DDE, and the age-structured models with matching parameter settings shows that the age structure promotes the instability of the Coexistence Equilibrium and the emergence of the Coexistence Periodic Attractor.

💡 Research Summary

The paper introduces a novel predator‑prey model that explicitly incorporates an age‑structured predator population together with functional role reversal (ontogenetic niche shift). Building on the ODE framework of Li, Liu, and Wei (2022), the authors replace the simplistic juvenile‑adult transition with a Kermack‑McKendrick‑type renewal equation for the predator age density (u(t,\tau)). Crucially, both the predator birth rate (B_p(x,\tau)) and death rate (\mu_p(x,\tau)) depend on the prey density (x(t)) and on whether the predator’s age (\tau) lies below or above a maturation threshold (\tau^\circ). Juvenile predators cannot reproduce and are vulnerable to predation by the prey, which is captured by a term proportional to a consumption rate (g). Adult predators hunt the prey, while all predators experience age‑related mortality and hunger‑induced mortality.

Mathematically, the authors prove existence, uniqueness, and positivity of solutions to the coupled ODE‑PDE initial‑value problem under standard boundedness and non‑negativity assumptions on the rate functions. The predator total population is obtained by integrating the age density over the juvenile ((0\le\tau<\tau^\circ)) and adult ((\tau\ge\tau^\circ)) intervals, yielding the variables (y_1(t)) and (y_2(t)) that feed back into the prey ODE.

To explore the high‑dimensional parameter space (15 parameters), the authors employ Latin Hypercube Sampling (LHS) to generate 10,000 uniformly distributed parameter sets, each simulated for a long horizon. The outcomes fall into four categories: (i) Equilibrium Coexistence Attractor (≈22 % of runs), (ii) Periodic Coexistence Attractor (≈19 %), (iii) Predator‑Free Attractor (≈55 %), and (iv) blow‑up (≈4 %). Linear Discriminant Analysis (LDA) identifies the maturation age (\tau^\circ) and the juvenile‑predator consumption rate (g) as the most influential parameters governing long‑term dynamics.

Focusing on the ((\tau^\circ,g)) plane, the authors construct detailed phase diagrams and bifurcation plots for two scenarios: a sharp versus a gradual maturation transition (controlled by a smooth indicator function with parameter (\nu)). They find that larger (\tau^\circ) and higher (g) expand the region where periodic coexistence emerges, while small values favor stable equilibrium or predator extinction. The age‑structured model also exhibits traveling‑wave‑like age‑density profiles during periodic cycles, contrasting with the monotone decay observed at equilibrium.

To assess the role of age structure per se, the authors derive two reduced models: (a) an ODE system tracking only juvenile and adult predator totals, and (b) a delay differential equation (DDE) that incorporates the maturation delay explicitly. All three models (age‑structured PDE, ODE, DDE) are simulated under identical parameter settings. The comparison reveals that the age‑structured PDE yields the largest parameter region supporting periodic coexistence, the ODE the smallest, and the DDE an intermediate size. This demonstrates that explicit age structure amplifies the system’s propensity for oscillatory dynamics, likely through the combination of distributed delays and nonlinear feedback introduced by the prey‑dependent birth/death rates.

The paper concludes with a discussion of ecological implications: age‑dependent role reversal can destabilize otherwise stable predator‑prey equilibria, potentially leading to cycles or predator collapse. The authors stress that management or conservation strategies should account for ontogenetic shifts, as interventions targeting only adult predators may miss critical dynamics driven by juveniles. All simulation code and analysis scripts are made publicly available on GitHub, facilitating reproducibility and future extensions to more complex food webs or spatially explicit settings.