Bifurcation analysis in a host-generalist parasitoid model with Holling II functional response

Chuang Xiang, Jicai Huang, Shigui Ruan, Dongmei Xiao

Research output: Contribution to journalArticle

Abstract

In this paper we study a host-generalist parasitoid model with Holling II functional response where the generalist parasitoids are introduced to control the invasion of the hosts. It is shown that the model can undergo a sequence of bifurcations including cusp, focus and elliptic types degenerate Bogdanov-Takens bifurcations of codimension three, and a degenerate Hopf bifurcation of codimension at most two as the parameters vary, and the model exhibits rich dynamics such as the existence of multiple coexistent steady states, multiple coexistent periodic orbits, homoclinic orbits, etc. Moreover, there exists a critical value for the carrying capacity of generalist parasitoids such that: (i) when the carrying capacity of the generalist parasitoids is smaller than the critical value, the invading hosts can always persist despite of the predation by the generalist parasitoids, i.e., the generalist parasitoids cannot control the invasion of hosts; (ii) when the carrying capacity of the generalist parasitoids is larger than the critical value, the invading hosts either tend to extinction or persist in the form of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations, i.e., whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populations; (iii) in both cases, the generalist parasitoids always persist. Numerical simulations are presented to illustrate the theoretical results.

Original languageEnglish (US)
JournalJournal of Differential Equations
DOIs
StateAccepted/In press - Jan 1 2019
Externally publishedYes

Fingerprint

Carrying Capacity
Functional Response
Invasion
Bifurcation Analysis
Critical value
Orbits
Periodic Orbits
Codimension
Bogdanov-Takens Bifurcation
Hopf bifurcation
Bifurcation (mathematics)
Homoclinic Orbit
Cusp
Extinction
Hopf Bifurcation
Bifurcation
Vary
Model
Tend
Numerical Simulation

Keywords

  • Bogdanov-Takens bifurcation
  • Extinction
  • Hopf bifurcation
  • Host-parasitoid model
  • Invasion
  • Persistence

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Bifurcation analysis in a host-generalist parasitoid model with Holling II functional response. / Xiang, Chuang; Huang, Jicai; Ruan, Shigui; Xiao, Dongmei.

In: Journal of Differential Equations, 01.01.2019.

Research output: Contribution to journalArticle

@article{e1986f3f4af1493dabe730e151d057e0,
title = "Bifurcation analysis in a host-generalist parasitoid model with Holling II functional response",
abstract = "In this paper we study a host-generalist parasitoid model with Holling II functional response where the generalist parasitoids are introduced to control the invasion of the hosts. It is shown that the model can undergo a sequence of bifurcations including cusp, focus and elliptic types degenerate Bogdanov-Takens bifurcations of codimension three, and a degenerate Hopf bifurcation of codimension at most two as the parameters vary, and the model exhibits rich dynamics such as the existence of multiple coexistent steady states, multiple coexistent periodic orbits, homoclinic orbits, etc. Moreover, there exists a critical value for the carrying capacity of generalist parasitoids such that: (i) when the carrying capacity of the generalist parasitoids is smaller than the critical value, the invading hosts can always persist despite of the predation by the generalist parasitoids, i.e., the generalist parasitoids cannot control the invasion of hosts; (ii) when the carrying capacity of the generalist parasitoids is larger than the critical value, the invading hosts either tend to extinction or persist in the form of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations, i.e., whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populations; (iii) in both cases, the generalist parasitoids always persist. Numerical simulations are presented to illustrate the theoretical results.",
keywords = "Bogdanov-Takens bifurcation, Extinction, Hopf bifurcation, Host-parasitoid model, Invasion, Persistence",
author = "Chuang Xiang and Jicai Huang and Shigui Ruan and Dongmei Xiao",
year = "2019",
month = "1",
day = "1",
doi = "10.1016/j.jde.2019.10.036",
language = "English (US)",
journal = "Journal of Differential Equations",
issn = "0022-0396",
publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Bifurcation analysis in a host-generalist parasitoid model with Holling II functional response

AU - Xiang, Chuang

AU - Huang, Jicai

AU - Ruan, Shigui

AU - Xiao, Dongmei

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In this paper we study a host-generalist parasitoid model with Holling II functional response where the generalist parasitoids are introduced to control the invasion of the hosts. It is shown that the model can undergo a sequence of bifurcations including cusp, focus and elliptic types degenerate Bogdanov-Takens bifurcations of codimension three, and a degenerate Hopf bifurcation of codimension at most two as the parameters vary, and the model exhibits rich dynamics such as the existence of multiple coexistent steady states, multiple coexistent periodic orbits, homoclinic orbits, etc. Moreover, there exists a critical value for the carrying capacity of generalist parasitoids such that: (i) when the carrying capacity of the generalist parasitoids is smaller than the critical value, the invading hosts can always persist despite of the predation by the generalist parasitoids, i.e., the generalist parasitoids cannot control the invasion of hosts; (ii) when the carrying capacity of the generalist parasitoids is larger than the critical value, the invading hosts either tend to extinction or persist in the form of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations, i.e., whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populations; (iii) in both cases, the generalist parasitoids always persist. Numerical simulations are presented to illustrate the theoretical results.

AB - In this paper we study a host-generalist parasitoid model with Holling II functional response where the generalist parasitoids are introduced to control the invasion of the hosts. It is shown that the model can undergo a sequence of bifurcations including cusp, focus and elliptic types degenerate Bogdanov-Takens bifurcations of codimension three, and a degenerate Hopf bifurcation of codimension at most two as the parameters vary, and the model exhibits rich dynamics such as the existence of multiple coexistent steady states, multiple coexistent periodic orbits, homoclinic orbits, etc. Moreover, there exists a critical value for the carrying capacity of generalist parasitoids such that: (i) when the carrying capacity of the generalist parasitoids is smaller than the critical value, the invading hosts can always persist despite of the predation by the generalist parasitoids, i.e., the generalist parasitoids cannot control the invasion of hosts; (ii) when the carrying capacity of the generalist parasitoids is larger than the critical value, the invading hosts either tend to extinction or persist in the form of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations, i.e., whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populations; (iii) in both cases, the generalist parasitoids always persist. Numerical simulations are presented to illustrate the theoretical results.

KW - Bogdanov-Takens bifurcation

KW - Extinction

KW - Hopf bifurcation

KW - Host-parasitoid model

KW - Invasion

KW - Persistence

UR - http://www.scopus.com/inward/record.url?scp=85074387750&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85074387750&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2019.10.036

DO - 10.1016/j.jde.2019.10.036

M3 - Article

AN - SCOPUS:85074387750

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -