Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A nonconservative kinetic framework with logistic growth for modeling the coexistence in a multi-species ecological system

Marco Menale, Carmelo Filippo Munafò, Francesco Oliveri

TL;DR

The paper introduces a nonconservative kinetic framework that incorporates a logistic growth–type external force field to model coexistence in a multi-species ecological system. By deriving mesoscopic evolution equations with an external force field $F_i[\mathbf{f}](t)$ and applying a two-species predator–prey interpretation, it reveals how open-system logistic growth interacts with binary interactions. The authors characterize the coexistence equilibrium $E_1=(f_1^\star,f_2^\star)$, analyze its linear stability across four scenarios, and demonstrate a Hopf bifurcation in a fourth scenario, supported by numerical simulations showing periodic orbits and basins of attraction. This framework extends classical Lotka-Volterra dynamics by explicitly modeling logistic growth effects via an environment-driven nonconservative term, with implications for open-system ecological dynamics and potential spatial extensions.

Abstract

Kinetic theory frameworks are widely used for modeling stochastic interacting systems, where the evolution primarily depends on binary interactions. Recently, in this framework the action of the external force field has been introduction in order to gain a more realistic picture of some phenomena. In this paper, we introduce nonconservative kinetic equations where a particular shape external force field acts on the overall system. Then, this framework is used in an ecological context for modeling the evolution of a system composed of two species interacting with a prey-predator mechanism. The linear stability analysis concerned with the coexistence equilibrium point is provided, and a case where a Hopf bifurcations occurs is discussed. Finally, some relevant scenarios are numerically simulated.

A nonconservative kinetic framework with logistic growth for modeling the coexistence in a multi-species ecological system

TL;DR

The paper introduces a nonconservative kinetic framework that incorporates a logistic growth–type external force field to model coexistence in a multi-species ecological system. By deriving mesoscopic evolution equations with an external force field and applying a two-species predator–prey interpretation, it reveals how open-system logistic growth interacts with binary interactions. The authors characterize the coexistence equilibrium , analyze its linear stability across four scenarios, and demonstrate a Hopf bifurcation in a fourth scenario, supported by numerical simulations showing periodic orbits and basins of attraction. This framework extends classical Lotka-Volterra dynamics by explicitly modeling logistic growth effects via an environment-driven nonconservative term, with implications for open-system ecological dynamics and potential spatial extensions.

Abstract

Kinetic theory frameworks are widely used for modeling stochastic interacting systems, where the evolution primarily depends on binary interactions. Recently, in this framework the action of the external force field has been introduction in order to gain a more realistic picture of some phenomena. In this paper, we introduce nonconservative kinetic equations where a particular shape external force field acts on the overall system. Then, this framework is used in an ecological context for modeling the evolution of a system composed of two species interacting with a prey-predator mechanism. The linear stability analysis concerned with the coexistence equilibrium point is provided, and a case where a Hopf bifurcations occurs is discussed. Finally, some relevant scenarios are numerically simulated.
Paper Structure (9 sections, 2 theorems, 43 equations, 4 figures)

This paper contains 9 sections, 2 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The kinetic model
  3. The ecological application
  4. The coexistence point: stability analysis
  5. First scenario
  6. Second scenario
  7. Third scenario
  8. Fourth scenario
  9. Conclusions

Key Result

Theorem 1

Let assume the kinetic framework eqnewfor. Along with the constraint assump, suppose that the three following further assumptions are satisfied: Then, there exists a unique solution $\mathbf{f}(t)=\left(f_1(t),\dots, f_n(t)\right)$ in the time interval $[0,\, t_0]$, that is positive and bounded.

Figures (4)

  • Figure 1: Time evolution of $f_1(t)$ and $f_2(t)$ in the time interval $[0,400]$, with different initial conditions (a) $\mathbf{f^0}=(0.0910,0.7450)$ and (b) $\mathbf{f^0}=(0.1150,0.7450)$. Subfigure (c) displays the phase portrait, and subfigure (d) the trajectories obtained by solving the system with different initial conditions chosen in the "basin of attraction" of $E_1$. The black dots represent the equilibria.
  • Figure 2: Time evolution of $f_1(t)$ and $f_2(t)$ in the time interval $[0,500]$, with different initial conditions (a) $\mathbf{f^0}=(0.0810,0.7450)$ and (b) $\mathbf{f^0}=(0.1060,0.7450)$. Subfigure (c) displays the phase portrait, and subfigure (d) the trajectories obtained by solving the system with different initial conditions chosen in the "basin of attraction" of $E_1$. The black dots represent the equilibria.
  • Figure 3: Time evolution of $f_1(t)$ and $f_2(t)$ in the time interval $[0,400]$, with different initial conditions (a) $\mathbf{f^0}=(0.1200,0.2500)$ and (b) $\mathbf{f^0}=(0.2000,0.3000)$. Subfigure (c) displays the phase portrait, and subfigure (d) the trajectories obtained by solving the system with different initial conditions chosen in the "basin of attraction" of $E_1$. The black dots represent the equilibria.
  • Figure 4: Time evolution of $f_1(t)$ and $f_2(t)$ in the time interval $[7000,7500]$, with different initial conditions (a) $\mathbf{f^0}=(0.9,0.9)$ and (b) $\mathbf{f^0}=(0.5,0.8)$. Subfigure (c) displays the phase portrait, and subfigure (d) the trajectories obtained by solving the system with different initial conditions chosen in the "basin of attraction" of $E_1$ (shown in subfigure (e)). The black dots represent the equilibria. It is possible to see that around the equilibria there are stable limit cycles.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof