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Exploring the refuge-induced bubbling phenomenon and harvesting in a three species food chain model that incorporates memory effect and odour effect

Dipam Das, Debasish Bhattacharjee

TL;DR

This work develops an odour-mediated three-species food chain model incorporating prey odour, predator-odour induced refuges, and prey harvesting, analyzed in both an ODE and a Caputo fractional-order framework to capture memory effects. It establishes well-posedness and derives four ecologically feasible equilibria ($E_v,E_a,E_t,E_c$), then conducts a rigorous bifurcation analysis showing transcritical and Hopf bifurcations and revealing a bubbling phenomenon in parameter space. Numerical simulations validate the analytical results and reveal how memory ($0<\alpha\le1$), refuge strength ($m_1,m_2$), and odour parameters ($a_1,a_2$, $\beta$) shape long-term dynamics, including sustained coexistence, oscillations, and extinctions. The findings underscore the ecological significance of odor cues and refuge behaviours in multi-species systems and offer insights for harvesting strategies and conservation under memory effects.

Abstract

In this study, an odour-mediated system is developed and studied. In an odor-mediated systems, the sense of smell or odour of species plays a critical role in the interactions between predators and prey. It is widely recognised in scientific literature that these systems are very common and essential across natural ecosystems. These systems are crucial for various behaviors, including foraging, mating, and avoiding predators. In this paper, it is assumed that the presence of prey odour aids the predator in its hunting efforts. It is assumed that both prey and intermediate predators seek refuge against their respective predators upon detecting the odour of their predators. In other words, the odour of predators assists prey species in evaluating the danger and seeking refuge for hiding. This model incorporates the prey species' harvesting as well. We also explore the impact of fading memory on the system by incorporating fractional derivatives into the model. The conditions for both the existence and local stability of the non-negative equilibria are derived. The current model system undergoes both Hopf and transcritical bifurcation when the parameter values are appropriately chosen. The dynamic behaviour of the system is showcased and thoroughly analysed using a range of diagrams, highlighting the impact of prey refuge and predator odour parameters. This paper extensively examines the long-term impacts of harvesting within the system. The extent to which prey odour influences the system is investigated, and it emerges that prey odour can play a significant function within the system. It has been observed that when individuals within the system have a strong memory, it positively affects the stability of the system. Numerical simulations are conducted in order to demonstrate and validate the usefulness of the model being considered, therefore supporting the analytical conclusions.

Exploring the refuge-induced bubbling phenomenon and harvesting in a three species food chain model that incorporates memory effect and odour effect

TL;DR

This work develops an odour-mediated three-species food chain model incorporating prey odour, predator-odour induced refuges, and prey harvesting, analyzed in both an ODE and a Caputo fractional-order framework to capture memory effects. It establishes well-posedness and derives four ecologically feasible equilibria (), then conducts a rigorous bifurcation analysis showing transcritical and Hopf bifurcations and revealing a bubbling phenomenon in parameter space. Numerical simulations validate the analytical results and reveal how memory (), refuge strength (), and odour parameters (, ) shape long-term dynamics, including sustained coexistence, oscillations, and extinctions. The findings underscore the ecological significance of odor cues and refuge behaviours in multi-species systems and offer insights for harvesting strategies and conservation under memory effects.

Abstract

In this study, an odour-mediated system is developed and studied. In an odor-mediated systems, the sense of smell or odour of species plays a critical role in the interactions between predators and prey. It is widely recognised in scientific literature that these systems are very common and essential across natural ecosystems. These systems are crucial for various behaviors, including foraging, mating, and avoiding predators. In this paper, it is assumed that the presence of prey odour aids the predator in its hunting efforts. It is assumed that both prey and intermediate predators seek refuge against their respective predators upon detecting the odour of their predators. In other words, the odour of predators assists prey species in evaluating the danger and seeking refuge for hiding. This model incorporates the prey species' harvesting as well. We also explore the impact of fading memory on the system by incorporating fractional derivatives into the model. The conditions for both the existence and local stability of the non-negative equilibria are derived. The current model system undergoes both Hopf and transcritical bifurcation when the parameter values are appropriately chosen. The dynamic behaviour of the system is showcased and thoroughly analysed using a range of diagrams, highlighting the impact of prey refuge and predator odour parameters. This paper extensively examines the long-term impacts of harvesting within the system. The extent to which prey odour influences the system is investigated, and it emerges that prey odour can play a significant function within the system. It has been observed that when individuals within the system have a strong memory, it positively affects the stability of the system. Numerical simulations are conducted in order to demonstrate and validate the usefulness of the model being considered, therefore supporting the analytical conclusions.

Paper Structure

This paper contains 21 sections, 14 theorems, 43 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Research contribution
  3. Model formulation
  4. Preliminaries
  5. Well-posedness of the systems
  6. Non-negativeness of solutions
  7. Boundedness of solutions
  8. Existence and uniqueness
  9. Existence of equilibrium points and their stability analysis
  10. Bifurcations
  11. Transcritical bifurcation
  12. Hopf bifurcation
  13. Numerical simulation
  14. Numerical verification of the local stability of the equilibrium points
  15. Effect of memory on the population dynamics (Impact of the parameter $\alpha$)
  16. ...and 6 more sections

Key Result

Lemma 1

kilb Let us assume, f(t) be a continuous function on (0,T] and satisfies $^{c}_{t_0}D_{t}^{\alpha}(f(t))\le - b_1 f(t)+ b_2$, $f(0)=f_{0}>0$, $0<\alpha<1$, where $b_1 \ne 0$ and $b_1,b_2\in \mathbb{R}$. Then Where, $E_{\alpha}$ is the Mittag–Leffler function.

Figures (18)

  • Figure 1: A flow chart illustrating the research methodology employed for the model under investigation in this scholarly article.
  • Figure 2: The graphical representation showcases the intricate relationships among the prey (grasshopper), intermediate predator (grasshopper mouse), and top predator (red fox).
  • Figure 3: This illustration demonstrates the local stability of all four ecologically feasible equilibrium points for different sets of parameter values.
  • Figure 4: This diagram illustrates the local stability of all the biologically viable fixed points of the system (\ref{['Frac eq']}) at a fractional order of $\alpha=0.98$.
  • Figure 5: This diagram illustrates the existence of various bifurcations in relation to different parameters in the system (\ref{['Final ode eq']}). The set of parameter values which are used to generate these figures are as follows: $r_1 = 2$, $r_5 = 1$, $\beta = 0.01$, $m_1 = 0.5$, $m_2 = 0.5$$d_1 = 0.25$, $r_2 = 1$, $d_2 = 0.5$, $r_4 = 3$, $b = 1$, $r_3 = 1$, $q = 0.5$, $r = 0.01$, and $\alpha=1$.
  • ...and 13 more figures

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 16 more