2007
DOI: 10.1080/17513750701605440
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A three-stage discrete-time population model: continuous versus seasonal reproduction

Abstract: We consider a three-stage discrete-time population model with density-dependent survivorship and time-dependent reproduction. We provide stability analysis for two types of birth mechanisms: continuous and seasonal. We show that when birth is continuous there exists a unique globally stable interior equilibrium provided that the inherent net reproductive number is greater than unity. If it is less than unity, then extinction is the population's fate. We then analyze the case when birth is a function of period … Show more

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Cited by 7 publications
(6 citation statements)
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References 15 publications
(23 reference statements)
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“…The main difficulty in studying the global attractivity of the positive fixed point for the full model (1) is that the nonlinear map governing the system is not monotone when 0 < γ 2 < 1. Thus, the theory developed for monotone systems (see for example, [14,20]) and used to study global attractivity of other stage-structured models (e.g., [1,2,17]) does not apply. Currently, no proof of global convergence to the unique positive fixed point is known in case γ 1 , γ 2 ∈ (0, 1] when the inherent net reproductive number R 0 (γ 1 , γ 2 ) is larger than one, although a large number of simulations we performed suggest that this might be the case.…”
Section: Discussionmentioning
confidence: 99%
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“…The main difficulty in studying the global attractivity of the positive fixed point for the full model (1) is that the nonlinear map governing the system is not monotone when 0 < γ 2 < 1. Thus, the theory developed for monotone systems (see for example, [14,20]) and used to study global attractivity of other stage-structured models (e.g., [1,2,17]) does not apply. Currently, no proof of global convergence to the unique positive fixed point is known in case γ 1 , γ 2 ∈ (0, 1] when the inherent net reproductive number R 0 (γ 1 , γ 2 ) is larger than one, although a large number of simulations we performed suggest that this might be the case.…”
Section: Discussionmentioning
confidence: 99%
“…Then Z is invariant for system (14), and we restrict the dynamics to Z henceforth. Define the continuous function P : Z → R + as follows 1]. Then clearly M is compact and invariant, and Z\M is positively invariant for (14).…”
Section: Where X(t) = (J (T) N (T) B(t)) and T γ (X(t)) Denotes Thmentioning
confidence: 99%
“…This assumption leads to a model which cannot be reduced to a single equation. Thus, we provide different arguments than those in [2,3] to study global stability. Furthermore, the Jacobian in this case is a full matrix.…”
Section: X(t + 1) = B(t)y(t) Y(t + 1) =mentioning
confidence: 99%
“…Here, it was demonstrated that unlike the two stage model, period-2 birth rates are not advantageous for low birth rates. The arguments in [2,3] rely on reducing the model to a higher order scalar difference equation and applying a result on monotone scalar difference equations from [19]. In general, the juvenile stage for some species is longer than the breeding season (e.g., the Bafo Boreas frog [29]).…”
Section: X(t + 1) = B(t)y(t) Y(t + 1) =mentioning
confidence: 99%
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