Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extinction, persistence and growing in a degenerate logistic model with impulses

Willian Cintra, Zhigui Lin, Carlos Alberto Santos, Phyu Phyu Win

Abstract

This paper deals with an impulsive degenerate logistic model, where pulses are introduced for modeling interventions or disturbances, and degenerate logistic term may describe refugees or protections zones for the species. Firstly, the principal eigenvalue depending on impulse rate, which is regarded as a threshold value, is introduced and characterized. Secondly, the asymptotic behavior of the population is fully investigated and sufficient conditions for the species to be extinct, persist or grow unlimitedly are given. Our results extend those of well-understood logistic and Malthusian models. Finally, numerical simulations emphanzise our theoretical results highlighting that medium impulse rate is more favorable for species to persist, small rate results in extinction and large rate leads the species to an unlimited growth.

Extinction, persistence and growing in a degenerate logistic model with impulses

Abstract

This paper deals with an impulsive degenerate logistic model, where pulses are introduced for modeling interventions or disturbances, and degenerate logistic term may describe refugees or protections zones for the species. Firstly, the principal eigenvalue depending on impulse rate, which is regarded as a threshold value, is introduced and characterized. Secondly, the asymptotic behavior of the population is fully investigated and sufficient conditions for the species to be extinct, persist or grow unlimitedly are given. Our results extend those of well-understood logistic and Malthusian models. Finally, numerical simulations emphanzise our theoretical results highlighting that medium impulse rate is more favorable for species to persist, small rate results in extinction and large rate leads the species to an unlimited growth.
Paper Structure (5 sections, 12 theorems, 75 equations, 3 figures)

This paper contains 5 sections, 12 theorems, 75 equations, 3 figures.

Table of Contents

  1. Introduction
  2. An eigenvalue problem with impulse condition
  3. About a impulsive periodic problem associated to Problem \ref{['a01']}
  4. The dynamical behavior of the solution
  5. Numerical simulation and biological explanation

Key Result

Lemma 2.1

Problem a013 admits a principal eigenvalue $\mu_1:=\mu_1(d, a(t,x), yb(t,x), z)$ with its corresponding eigenfunction $\phi(t,x) \gg 0$ in $[0,\tau] \times \Omega$. Moreover, $\mu_1(d, a(t,x), yb(t,x), z)$ is increasing in $d>0$ and $y\geq 0$.

Figures (3)

  • Figure 1: The dynamics of species $u$ with small pulse $c=0.08$ at every $\tau=2$. Graph $(a)$ is the right view of the spatiotemporal distribution of $u$ plotted in graph $(b)$, while in the Graph $(c)$ are plotted the cross sections view at $t=0,3,5,7, 20$. Graphs $(a)-(c)$ imply that the species $u(t,x)$ gradually tends to zero with time $t$.
  • Figure 2: The dynamics of species $u$ with medium pulse $c=0.8$ at every $\tau=2$. Graph $(a)$ is the right view of the spatiotemporal distribution of $u$ in the graph $(b)$, and Graph $(c)$ is its cross section views at $t=0,3,5,7, 20$. Graphs $(a)- (c)$ imply that the species $u(t,x)$ gradually tends to a periodic state.
  • Figure 3: The dynamics of species $u$ with large pulse $c=0.08$ at every $\tau=2$. Graph $(a)$ is the right view of the spatiotemporal distribution of $u$ in graph $(b)$, and Graph $(c)$ is its cross section views at $t=0,3,5,7, 20$. Graphs $(a)- (c)$ imply that the species $u(t,x)$ increases quickly.

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.1
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Theorem 4.1
  • ...and 3 more