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An eco-epidemiological model with prey-taxis and slow diffusion: Global existence, boundedness and novel dynamics

Ranjit Kumar Upadhyay, Rana D. Parshad, Namrata Mani Tripathi, Nishith Mohan

TL;DR

This work analyzes an eco-epidemiological model that couples predator-prey interactions with prey-taxis and slow diffusion of infected prey through a $p$-Laplacian, incorporating nonlinear mortality $-\mu I^{\gamma}$. It establishes global well-posedness in two regimes: (i) classical diffusion with linear mortality ($p=2$, $\gamma=1$) yielding global classical solutions, and (ii) slow diffusion with nonlinear mortality ($p>2$, $0<\gamma<1$) yielding global weak solutions, with uniform bounds and convergence results. The study then characterizes stability and bifurcation phenomena for the linear-diffusion case, proving steady-state and Hopf bifurcation curves with explicit expressions for critical taxis parameters, and demonstrating that taxis can induce or destroy pattern formation. Numerical simulations corroborate the analytical insights, showing taxis-driven transition from a stable interior equilibrium to spatial patterns or temporal oscillations; finite-time extinction phenomena are also established in the presence of nonlinear absorption. Overall, the results illuminate how prey-taxis and slow diffusion shape global dynamics, with implications for biological invasions and pest management under disease in prey populations.

Abstract

In this manuscript, an attempt has been made to understand the effects of prey-taxis on the existence of global-in-time solutions and dynamics in an eco-epidemiological model, particularly under the influence of slow dispersal characterized by the $p$-Laplacian operator and enhanced mortality of the infected prey, subject to specific assumptions on the taxis sensitivity functions. We prove the global existence of classical solutions when the infected prey undergoes random motion and exhibits standard mortality. Under the assumption that the infected prey disperses slowly and exhibits enhanced mortality, we prove the global existence of weak solutions. Following a detailed mathematical investigation of the proposed model, we shift our focus to analyse the stability of the positive equilibrium point under the scenario where all species exhibit linear diffusion, the infected prey experiences standard mortality, and the predator exhibits taxis exclusively toward the infected prey. Within this framework, we establish the occurrence of a steady-state bifurcation. Numerical simulations are then carried out to observe this dynamical behavior. Our results have large scale applications to biological invasions and biological control of pests, under the prevalence of disease in the pest population.

An eco-epidemiological model with prey-taxis and slow diffusion: Global existence, boundedness and novel dynamics

TL;DR

This work analyzes an eco-epidemiological model that couples predator-prey interactions with prey-taxis and slow diffusion of infected prey through a -Laplacian, incorporating nonlinear mortality . It establishes global well-posedness in two regimes: (i) classical diffusion with linear mortality (, ) yielding global classical solutions, and (ii) slow diffusion with nonlinear mortality (, ) yielding global weak solutions, with uniform bounds and convergence results. The study then characterizes stability and bifurcation phenomena for the linear-diffusion case, proving steady-state and Hopf bifurcation curves with explicit expressions for critical taxis parameters, and demonstrating that taxis can induce or destroy pattern formation. Numerical simulations corroborate the analytical insights, showing taxis-driven transition from a stable interior equilibrium to spatial patterns or temporal oscillations; finite-time extinction phenomena are also established in the presence of nonlinear absorption. Overall, the results illuminate how prey-taxis and slow diffusion shape global dynamics, with implications for biological invasions and pest management under disease in prey populations.

Abstract

In this manuscript, an attempt has been made to understand the effects of prey-taxis on the existence of global-in-time solutions and dynamics in an eco-epidemiological model, particularly under the influence of slow dispersal characterized by the -Laplacian operator and enhanced mortality of the infected prey, subject to specific assumptions on the taxis sensitivity functions. We prove the global existence of classical solutions when the infected prey undergoes random motion and exhibits standard mortality. Under the assumption that the infected prey disperses slowly and exhibits enhanced mortality, we prove the global existence of weak solutions. Following a detailed mathematical investigation of the proposed model, we shift our focus to analyse the stability of the positive equilibrium point under the scenario where all species exhibit linear diffusion, the infected prey experiences standard mortality, and the predator exhibits taxis exclusively toward the infected prey. Within this framework, we establish the occurrence of a steady-state bifurcation. Numerical simulations are then carried out to observe this dynamical behavior. Our results have large scale applications to biological invasions and biological control of pests, under the prevalence of disease in the pest population.

Paper Structure

This paper contains 16 sections, 25 theorems, 209 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Boundedness and Global Existence
  3. The case $p=2, \gamma=1$
  4. The case $p>2, 0<\gamma<1$
  5. Preliminaries and $L^{q}(\Omega)$ bounds
  6. Alikakos-Moser iteration
  7. Global Existence of Weak Solutions
  8. Convergence of the Non-linear gradient term
  9. Stability analysis of the positive equilibrium ($p=2, \gamma =1$ case)
  10. Steady state bifurcation
  11. Stability analysis of boundary equilibrium - Finite time extinction
  12. The $p=2,1>\gamma>0$ case
  13. The $p>2,0<\gamma<1$ case
  14. Thresholds for disease spread in temporal case with $0<\gamma <1$
  15. Numerical Simulations
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\Omega \subset \mathbb{R}^n (n \geq 1)$ be a bounded domain with a smooth boundary $\partial \Omega$. Then if the initial conditions are such that $(S_0, I_0, P_0) \in (W^{1, p}(\Omega))^3$ with $S_0, I_0, P_0 \geq 0$ and $p > n$ holds true and Assumption assu2 holds true. Then there exists a g

Figures (5)

  • Figure 1: $\mathcal{E}^*$ remains stable for \ref{['eq:model1']} when $\chi_1=\chi_2=0$
  • Figure 2: $\mathcal{E}^*$ remains stable for \ref{['eq:model1']} for small taxis $\chi_1=\chi_2=10$
  • Figure 3: Instability of $\mathcal{E}^*$ induced by large taxis $\chi_1=\chi_2=100$
  • Figure 4: Induced periodicity with some fluctuation when $\chi_1=100, \chi_2=40$
  • Figure 5: Irregular patterns occur when $\chi_1=40, \chi_2=100$

Theorems & Definitions (50)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['thm:main_result']}
  • Definition 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • ...and 40 more