### Abstract

We consider discrete time linear population models of the form n (t + 1) = An (t) where A is a population projection matrix or integral projection operator, and n (t) represents a structured population at time t. It is well known that the asymptotic growth or decay rate of n (t) is determined by the leading eigenvalue of A. In practice, population models have substantial parameter uncertainty, and it might be difficult to quantify the effect of this uncertainty on the leading eigenvalue. For a large class of matrices and integral operators A, we give sufficient conditions for an eigenvalue to be the leading eigenvalue. By preselecting the leading eigenvalue to be equal to 1, this allows us to easily identify, which combination of parameters, within the confines of their uncertainty, lead to asymptotic growth, and which lead to asymptotic decay. We then apply these results to the analysis of uncertainty in both a matrix model and an integral model for a population of thistles. We show these results can be generalized to any preselected leading eigenvalue.

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

Pages (from-to) | 85-97 |

Number of pages | 13 |

Journal | Theoretical Population Biology |

Volume | 75 |

Issue number | 2-3 |

DOIs | |

State | Published - Mar 1 2009 |

### Fingerprint

### Keywords

- Asymptotic growth rate
- Integral projection model
- Population projection matrix
- Robustness

### ASJC Scopus subject areas

- Ecology, Evolution, Behavior and Systematics

### Cite this

*Theoretical Population Biology*,

*75*(2-3), 85-97. https://doi.org/10.1016/j.tpb.2008.11.004

**Parameterizing the growth-decline boundary for uncertain population projection models.** / Lubben, Joan; Boeckner, Derek; Rebarber, Richard; Townley, Stuart; Tenhumberg, Brigitte.

Research output: Contribution to journal › Article

*Theoretical Population Biology*, vol. 75, no. 2-3, pp. 85-97. https://doi.org/10.1016/j.tpb.2008.11.004

}

TY - JOUR

T1 - Parameterizing the growth-decline boundary for uncertain population projection models

AU - Lubben, Joan

AU - Boeckner, Derek

AU - Rebarber, Richard

AU - Townley, Stuart

AU - Tenhumberg, Brigitte

PY - 2009/3/1

Y1 - 2009/3/1

N2 - We consider discrete time linear population models of the form n (t + 1) = An (t) where A is a population projection matrix or integral projection operator, and n (t) represents a structured population at time t. It is well known that the asymptotic growth or decay rate of n (t) is determined by the leading eigenvalue of A. In practice, population models have substantial parameter uncertainty, and it might be difficult to quantify the effect of this uncertainty on the leading eigenvalue. For a large class of matrices and integral operators A, we give sufficient conditions for an eigenvalue to be the leading eigenvalue. By preselecting the leading eigenvalue to be equal to 1, this allows us to easily identify, which combination of parameters, within the confines of their uncertainty, lead to asymptotic growth, and which lead to asymptotic decay. We then apply these results to the analysis of uncertainty in both a matrix model and an integral model for a population of thistles. We show these results can be generalized to any preselected leading eigenvalue.

AB - We consider discrete time linear population models of the form n (t + 1) = An (t) where A is a population projection matrix or integral projection operator, and n (t) represents a structured population at time t. It is well known that the asymptotic growth or decay rate of n (t) is determined by the leading eigenvalue of A. In practice, population models have substantial parameter uncertainty, and it might be difficult to quantify the effect of this uncertainty on the leading eigenvalue. For a large class of matrices and integral operators A, we give sufficient conditions for an eigenvalue to be the leading eigenvalue. By preselecting the leading eigenvalue to be equal to 1, this allows us to easily identify, which combination of parameters, within the confines of their uncertainty, lead to asymptotic growth, and which lead to asymptotic decay. We then apply these results to the analysis of uncertainty in both a matrix model and an integral model for a population of thistles. We show these results can be generalized to any preselected leading eigenvalue.

KW - Asymptotic growth rate

KW - Integral projection model

KW - Population projection matrix

KW - Robustness

UR - http://www.scopus.com/inward/record.url?scp=64549158400&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=64549158400&partnerID=8YFLogxK

U2 - 10.1016/j.tpb.2008.11.004

DO - 10.1016/j.tpb.2008.11.004

M3 - Article

C2 - 19105968

AN - SCOPUS:64549158400

VL - 75

SP - 85

EP - 97

JO - Theoretical Population Biology

JF - Theoretical Population Biology

SN - 0040-5809

IS - 2-3

ER -