### Abstract

In the presence of seasonal forcing, predator-prey models with quadratic interaction terms and weak dissipation can exhibit infinite numbers of coexisting periodic attractors corresponding to cycles of different magnitude and frequency. These motions are best understood with reference to the conservative case, for which the degree of dissipation is, by definition, zero. Here one observes the familiar mix of 'regular' (neutrally stable orbits and tori) and chaotic motion typical of non-integrable Hamiltonian systems. Perturbing away from the conservative limit, the chaos becomes transitory. In addition, the invariant tori are destroyed and the neutrally stable periodic orbits becomes stable limit cycles, the basins of attraction of which are intertwined in a complicated fashion. As a result, stochastic perturbations can bounce the system from one basin to another with consequent changes in system behavior. Biologically, weak dissipation corresponds to the case in which predators are able to regulate the density of their prey well below carrying capacity.

Original language | English (US) |
---|---|

Pages (from-to) | 835-859 |

Number of pages | 25 |

Journal | Bulletin of Mathematical Biology |

Volume | 58 |

Issue number | 5 |

State | Published - Sep 1996 |

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### ASJC Scopus subject areas

- Agricultural and Biological Sciences(all)

### Cite this

*Bulletin of Mathematical Biology*,

*58*(5), 835-859.

**Weakly dissipative predator-prey systems.** / King, A. A.; Schaffer, William M; Gordon, C.; Treat, J.; Kot, M.

Research output: Contribution to journal › Article

*Bulletin of Mathematical Biology*, vol. 58, no. 5, pp. 835-859.

}

TY - JOUR

T1 - Weakly dissipative predator-prey systems

AU - King, A. A.

AU - Schaffer, William M

AU - Gordon, C.

AU - Treat, J.

AU - Kot, M.

PY - 1996/9

Y1 - 1996/9

N2 - In the presence of seasonal forcing, predator-prey models with quadratic interaction terms and weak dissipation can exhibit infinite numbers of coexisting periodic attractors corresponding to cycles of different magnitude and frequency. These motions are best understood with reference to the conservative case, for which the degree of dissipation is, by definition, zero. Here one observes the familiar mix of 'regular' (neutrally stable orbits and tori) and chaotic motion typical of non-integrable Hamiltonian systems. Perturbing away from the conservative limit, the chaos becomes transitory. In addition, the invariant tori are destroyed and the neutrally stable periodic orbits becomes stable limit cycles, the basins of attraction of which are intertwined in a complicated fashion. As a result, stochastic perturbations can bounce the system from one basin to another with consequent changes in system behavior. Biologically, weak dissipation corresponds to the case in which predators are able to regulate the density of their prey well below carrying capacity.

AB - In the presence of seasonal forcing, predator-prey models with quadratic interaction terms and weak dissipation can exhibit infinite numbers of coexisting periodic attractors corresponding to cycles of different magnitude and frequency. These motions are best understood with reference to the conservative case, for which the degree of dissipation is, by definition, zero. Here one observes the familiar mix of 'regular' (neutrally stable orbits and tori) and chaotic motion typical of non-integrable Hamiltonian systems. Perturbing away from the conservative limit, the chaos becomes transitory. In addition, the invariant tori are destroyed and the neutrally stable periodic orbits becomes stable limit cycles, the basins of attraction of which are intertwined in a complicated fashion. As a result, stochastic perturbations can bounce the system from one basin to another with consequent changes in system behavior. Biologically, weak dissipation corresponds to the case in which predators are able to regulate the density of their prey well below carrying capacity.

UR - http://www.scopus.com/inward/record.url?scp=0030237581&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030237581&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030237581

VL - 58

SP - 835

EP - 859

JO - Bulletin of Mathematical Biology

JF - Bulletin of Mathematical Biology

SN - 0092-8240

IS - 5

ER -