Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A predator-prey model with age-structured role reversal

Luis Suarez, Maria K. Cameron, William F. Fagan, Doron Levy

TL;DR

The paper addresses predator-prey dynamics with an age-structured predator population that exhibits ontogenetic role reversal, formulated via a PDE-based renewal equation whose birth and death rates depend on prey size. It develops a mixed PDE/ODE framework with a maturation age $\tau^{*}$, proves existence, uniqueness, and positivity of solutions, and derives ODE and DDE reductions to enable comparison. Through Latin Hypercube Sampling of 15 parameters and Linear Discriminant Analysis, the study identifies $\tau^{*}$ and juvenile-predation rate $g$ as the most influential factors shaping long-term outcomes, including Predator-Free, Equilibrial Coexistence, and Periodic Coexistence attractors, with occasional blow-up in the un-saturated version. The comparisons show that the age-structured model more readily yields oscillatory dynamics than the ODE and DDE approximations, underscoring the importance of preserving age structure for ecological realism and cautioning against overreliance on reduced models.

Abstract

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.

A predator-prey model with age-structured role reversal

TL;DR

The paper addresses predator-prey dynamics with an age-structured predator population that exhibits ontogenetic role reversal, formulated via a PDE-based renewal equation whose birth and death rates depend on prey size. It develops a mixed PDE/ODE framework with a maturation age , proves existence, uniqueness, and positivity of solutions, and derives ODE and DDE reductions to enable comparison. Through Latin Hypercube Sampling of 15 parameters and Linear Discriminant Analysis, the study identifies and juvenile-predation rate as the most influential factors shaping long-term outcomes, including Predator-Free, Equilibrial Coexistence, and Periodic Coexistence attractors, with occasional blow-up in the un-saturated version. The comparisons show that the age-structured model more readily yields oscillatory dynamics than the ODE and DDE approximations, underscoring the importance of preserving age structure for ecological realism and cautioning against overreliance on reduced models.

Abstract

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.

Paper Structure

This paper contains 35 sections, 9 theorems, 106 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. An overview
  3. Mathematical modeling of ecosystems
  4. The goal and a summary of main results
  5. Model development
  6. A building block: Li, Liu, and Wei's ODE
  7. The Kermack-McKendrick Renewal Equation
  8. An age-structured predator-prey PDE-based model
  9. Existence, uniqueness, and positivity
  10. Characteristics of the predator equation
  11. Discretization
  12. The main theorem
  13. Reduction to an ODE model
  14. Reduction to a delayed differential equation
  15. Numerics
  16. ...and 20 more sections

Key Result

Theorem 1

Let the initial prey population size $x_0$ be positive. Let the initial predator age density $u_0(x)$ be nonnegative, piecewise smooth, and compactly supported on $[0,\tau_{0,\max}]$. Let the constants $r$, $a$, $s$, $b$, and $\mu_M$ be positive and the constants $k$ and $g$ be nonnegative. Furtherm

Figures (12)

  • Figure 1: The age-time diagram for the model \ref{['eq:model-prey']}--\ref{['eq:smooth_phi']}.
  • Figure 2: The results of Latin Hypercube Sampling. The intervals for the parameters are specified in Table \ref{['table2']}.
  • Figure 3: LDA projection from the 15D parameter space onto a 2D optimal subspace where the parameter sets leading to four different types of long-term behavior (Blow-Up, Coexistence Equilibrium, Coexistence periodic, and Predator-free attractor) are separated the most.
  • Figure 4: LDA range-adjusted bar graphs quantifying the importance of the model parameters for the type of long-term behavior.
  • Figure 5: The phase diagram in the $(\tau^*,g)$-plane and three bifurcation diagrams for the proposed model \ref{['eq:model-prey']}--\ref{['eq:smooth_phi']}. The rest of the parameters are at their selected values -- see Table \ref{['table2']}.
  • ...and 7 more figures

Theorems & Definitions (18)

  • Theorem 1
  • Remark
  • Lemma A1
  • proof
  • Lemma A2
  • proof
  • Lemma A3
  • proof
  • Lemma A4
  • proof
  • ...and 8 more