Stabilization and complex dynamics in a predator-prey model with predator suffering from an infectious disease.

B.W. Kooi, G.A.K. van Voorn, K. Pada Das

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We study the effects of a non-specified infectious disease of the predator on the dynamics a predator-prey system, by evaluating the dynamics of a three-dimensional model. The predator population in this (PSI) model is split into a susceptible and an unrecoverable infected population, while all newborn are susceptible. The incidence rate at which susceptible become infectious is described by a Holling type II functional response giving saturation when the number of susceptibles increases. From a modeling context this three-dimensional model is in the limit case similar to the well-known 3D Rosenzweig-MacArthur (RM) model, with the infected population replacing the top-predator. The RM model is known for the Shil'nikov bifurcation, which is associated to the chaotic behaviour. The effects of the disease are considered to be changes in the parameters that represent relative predation efficiency and mortality rates. A combination of analysis, numerical integration and numerical continuation techniques are used to perform a bifurcation analysis of the model. The positive stationary solution of the disease free, two-dimensional predator-prey system is either a stable equilibrium or a stable limit cycle where the transition occurs at the Hopf bifurcation. For a biologically applicable parameter set, it is found that when the infected individuals feed less fast or less effective than the susceptibles there is bi-stability where the two-dimensional disease free state co-exists with a stable equilibrium for the three-dimensional PSI system. The introduction of a disease can also cause chaos when the infected predator individuals are ecologically not functioning (not feeding and no offspring). However, under small parameter changes first the Shil'nikov bifurcation, and hence the chaotic behaviour, disappears followed by the Hopf bifurcation that marks the existence of limit cycles of the three-dimensional PSI system. As such, an infectious disease has a strongly stabilizing effect on the predator-prey system, similar to the existence of weak links in food webs. © 2010 Elsevier B.V.
Original languageEnglish
Pages (from-to)113-122
JournalEcological Complexity
Volume8
DOIs
Publication statusPublished - 2011

Fingerprint

infectious disease
infectious diseases
stabilization
bifurcation
predator
predators
noninfectious diseases
functional response
functional response models
chaotic dynamics
food webs
food web
neonates
predation
saturation
incidence
mortality
modeling
parameter
effect

Cite this

@article{23e5074ca64549108303d26b5cf5fec7,
title = "Stabilization and complex dynamics in a predator-prey model with predator suffering from an infectious disease.",
abstract = "We study the effects of a non-specified infectious disease of the predator on the dynamics a predator-prey system, by evaluating the dynamics of a three-dimensional model. The predator population in this (PSI) model is split into a susceptible and an unrecoverable infected population, while all newborn are susceptible. The incidence rate at which susceptible become infectious is described by a Holling type II functional response giving saturation when the number of susceptibles increases. From a modeling context this three-dimensional model is in the limit case similar to the well-known 3D Rosenzweig-MacArthur (RM) model, with the infected population replacing the top-predator. The RM model is known for the Shil'nikov bifurcation, which is associated to the chaotic behaviour. The effects of the disease are considered to be changes in the parameters that represent relative predation efficiency and mortality rates. A combination of analysis, numerical integration and numerical continuation techniques are used to perform a bifurcation analysis of the model. The positive stationary solution of the disease free, two-dimensional predator-prey system is either a stable equilibrium or a stable limit cycle where the transition occurs at the Hopf bifurcation. For a biologically applicable parameter set, it is found that when the infected individuals feed less fast or less effective than the susceptibles there is bi-stability where the two-dimensional disease free state co-exists with a stable equilibrium for the three-dimensional PSI system. The introduction of a disease can also cause chaos when the infected predator individuals are ecologically not functioning (not feeding and no offspring). However, under small parameter changes first the Shil'nikov bifurcation, and hence the chaotic behaviour, disappears followed by the Hopf bifurcation that marks the existence of limit cycles of the three-dimensional PSI system. As such, an infectious disease has a strongly stabilizing effect on the predator-prey system, similar to the existence of weak links in food webs. {\circledC} 2010 Elsevier B.V.",
author = "B.W. Kooi and {van Voorn}, G.A.K. and {Pada Das}, K.",
year = "2011",
doi = "10.1016/j.ecocom.2010.11.002",
language = "English",
volume = "8",
pages = "113--122",
journal = "Ecological Complexity",
issn = "1476-945X",
publisher = "Elsevier",

}

Stabilization and complex dynamics in a predator-prey model with predator suffering from an infectious disease. / Kooi, B.W.; van Voorn, G.A.K.; Pada Das, K.

In: Ecological Complexity, Vol. 8, 2011, p. 113-122.

Research output: Contribution to JournalArticleAcademicpeer-review

TY - JOUR

T1 - Stabilization and complex dynamics in a predator-prey model with predator suffering from an infectious disease.

AU - Kooi, B.W.

AU - van Voorn, G.A.K.

AU - Pada Das, K.

PY - 2011

Y1 - 2011

N2 - We study the effects of a non-specified infectious disease of the predator on the dynamics a predator-prey system, by evaluating the dynamics of a three-dimensional model. The predator population in this (PSI) model is split into a susceptible and an unrecoverable infected population, while all newborn are susceptible. The incidence rate at which susceptible become infectious is described by a Holling type II functional response giving saturation when the number of susceptibles increases. From a modeling context this three-dimensional model is in the limit case similar to the well-known 3D Rosenzweig-MacArthur (RM) model, with the infected population replacing the top-predator. The RM model is known for the Shil'nikov bifurcation, which is associated to the chaotic behaviour. The effects of the disease are considered to be changes in the parameters that represent relative predation efficiency and mortality rates. A combination of analysis, numerical integration and numerical continuation techniques are used to perform a bifurcation analysis of the model. The positive stationary solution of the disease free, two-dimensional predator-prey system is either a stable equilibrium or a stable limit cycle where the transition occurs at the Hopf bifurcation. For a biologically applicable parameter set, it is found that when the infected individuals feed less fast or less effective than the susceptibles there is bi-stability where the two-dimensional disease free state co-exists with a stable equilibrium for the three-dimensional PSI system. The introduction of a disease can also cause chaos when the infected predator individuals are ecologically not functioning (not feeding and no offspring). However, under small parameter changes first the Shil'nikov bifurcation, and hence the chaotic behaviour, disappears followed by the Hopf bifurcation that marks the existence of limit cycles of the three-dimensional PSI system. As such, an infectious disease has a strongly stabilizing effect on the predator-prey system, similar to the existence of weak links in food webs. © 2010 Elsevier B.V.

AB - We study the effects of a non-specified infectious disease of the predator on the dynamics a predator-prey system, by evaluating the dynamics of a three-dimensional model. The predator population in this (PSI) model is split into a susceptible and an unrecoverable infected population, while all newborn are susceptible. The incidence rate at which susceptible become infectious is described by a Holling type II functional response giving saturation when the number of susceptibles increases. From a modeling context this three-dimensional model is in the limit case similar to the well-known 3D Rosenzweig-MacArthur (RM) model, with the infected population replacing the top-predator. The RM model is known for the Shil'nikov bifurcation, which is associated to the chaotic behaviour. The effects of the disease are considered to be changes in the parameters that represent relative predation efficiency and mortality rates. A combination of analysis, numerical integration and numerical continuation techniques are used to perform a bifurcation analysis of the model. The positive stationary solution of the disease free, two-dimensional predator-prey system is either a stable equilibrium or a stable limit cycle where the transition occurs at the Hopf bifurcation. For a biologically applicable parameter set, it is found that when the infected individuals feed less fast or less effective than the susceptibles there is bi-stability where the two-dimensional disease free state co-exists with a stable equilibrium for the three-dimensional PSI system. The introduction of a disease can also cause chaos when the infected predator individuals are ecologically not functioning (not feeding and no offspring). However, under small parameter changes first the Shil'nikov bifurcation, and hence the chaotic behaviour, disappears followed by the Hopf bifurcation that marks the existence of limit cycles of the three-dimensional PSI system. As such, an infectious disease has a strongly stabilizing effect on the predator-prey system, similar to the existence of weak links in food webs. © 2010 Elsevier B.V.

U2 - 10.1016/j.ecocom.2010.11.002

DO - 10.1016/j.ecocom.2010.11.002

M3 - Article

VL - 8

SP - 113

EP - 122

JO - Ecological Complexity

JF - Ecological Complexity

SN - 1476-945X

ER -