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Population size might be regulated by competition for suitable territories. Consider a large number of suitable territories or patches. At time t, a fraction p(t) of these patches are occupied. Of the unoccupied sites, a fraction mp(t) are recolonized from occupied patches. Subsequently, each occupied site suffers a risk of local extinction e through catastrophic events such as fire or disease. These assumptions are consistent with the following differential equation:

Population size might be regulated by competition for suitable territories. Consider a large number of suitable territories or patches. At time t, a fraction p(t) of these patches are occupied. Of the unoccupied sites, a fraction mp(t) are recolonized from occupied patches. Subsequently, each occupied site suffers a risk of local extinction e through catastrophic events such as fire or disease. These assumptions are consistent with the following differential equation:

Applications and Investigations in Earth Science (9th Edition)
Applications and Investigations in Earth Science (9th Edition)
9th Edition
ISBN:9780134746241
Author:Edward J. Tarbuck, Frederick K. Lutgens, Dennis G. Tasa
Publisher:Edward J. Tarbuck, Frederick K. Lutgens, Dennis G. Tasa
Chapter1: The Study Of Minerals
Section: Chapter Questions
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Draw phase diagrams

Population size might be regulated by competition for suitable territories. Consider a large number of suitable territories or patches. At time t, a fraction p(t) of these patches are occupied. Of the unoccupied sites, a fraction mp(t) are recolonized from occupied patches. Subsequently, each occupied site suffers a risk of local extinction e through catastrophic events such as fire or disease. These assumptions are consistent with the following differential equation: to Find the equili dp dt (a) Find the two equilibria of this model. (b) Under what conditions is there a biologically valid equilibrium with the species present? set &f=0 / de = mp(1-p) - ep. (c) Given that the equilibrium in (b) is valid, when is it stable? (d) Is it possible for the fraction of occupied sites to overshoot the equilibrium? (Hint: how can you relate the phase portraits to the trajectories of the solution?)
Transcribed Image Text:Population size might be regulated by competition for suitable territories. Consider a large number of suitable territories or patches. At time t, a fraction p(t) of these patches are occupied. Of the unoccupied sites, a fraction mp(t) are recolonized from occupied patches. Subsequently, each occupied site suffers a risk of local extinction e through catastrophic events such as fire or disease. These assumptions are consistent with the following differential equation: to Find the equili dp dt (a) Find the two equilibria of this model. (b) Under what conditions is there a biologically valid equilibrium with the species present? set &f=0 / de = mp(1-p) - ep. (c) Given that the equilibrium in (b) is valid, when is it stable? (d) Is it possible for the fraction of occupied sites to overshoot the equilibrium? (Hint: how can you relate the phase portraits to the trajectories of the solution?)
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