Discrete three-stage population model: persistence and global stability results

Chiquet

2009

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A general discrete juvenile-adult population model with time-dependent birth rate and nonlinear survivorship rates is studied. When breeding is continuous, it is shown that the model has a unique globally asymptotically stable positive equilibrium provided the net reproductive number is larger than one. If it is smaller than one, then the extinction equilibrium is globally asymptotically stable. When breeding is seasonal, it is shown that there exists a unique globally asymptotically stable periodic solution provided the net reproductive number is larger than one. When this value is less than one, the population goes to extinction. Conditions on the birth rate where the population with seasonal breeding survives while the population with continuous breeding becomes extinct are provided.

“…In the first case, we assume b(t) = b, a positive constant, and in the second case, we assume b(t) is periodic with period 2. The following result follows from [1]. Here, P : R …”

Section: Two-stage Discrete Modelmentioning

confidence: 64%

“…Thus (2.6) holds, and E 1 is locally asymptotically stable. Now we will establish global attractivity of E 1 by following an approach similar to that in [1]. Since DP (x) is a nonnegative matrix for all x, we have that P (x) is monotone.…”

Section: Using This We Getmentioning

confidence: 99%

The global dynamics of a discrete juvenile–adult model with continuous and seasonal reproduction

Chiquet

2009

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“…This facilitates the proof of the global attractivity of the interior equilibrium [2,19], which is given in the next theorem. + to be the right side of system (1).…”

Section: Constant Birth and Dispersionmentioning

confidence: 99%

“…Proof We will use techniques similar to [2,19]. All vector and matrix inequalities hold componentwise.…”

Section: Constant Birth and Dispersionmentioning

confidence: 99%

A discrete dispersal model with constant and periodic environments

Chiquet

Zhang

2011

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We study a discrete juvenile-adult model which describes the dynamics of a population that reproduces and disperses between two patches constantly or seasonally. When breeding and dispersal rates are constant, the model has a unique interior equilibrium that is globally attractive, provided the net reproductive number is greater than one. If net reproductive number is less than one, then the extinction equilibrium is globally asymptotically stable. When breeding and dispersal rates are periodic of period 2, the extinction equilibrium is globally asymptotically stable if the net reproductive number is less than one. If the net reproductive number is greater than one, then there exists a unique globally attractive periodic solution. We then use bifurcation analysis to compare constant and seasonal breeding strategies, to explore the effects of different birth and dispersal periodicities and to understand the influence of strong nonlinearities on the dynamics of the model.

“…To this end, we point out that by revisiting the proof of Lemma 3 in [1] one can see that this lemma holds for the competitive …”

Section: H(t (X)) = H((mmentioning

confidence: 99%

A discrete stage-structured two-species competition model with sexual and clonal reproduction

Zhang

2012

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In this paper, we analyse a discrete stage-structured model which is a generalization of the two-species competition model studied in [2]. Motivated by plant populations, each species is assumed to reproduce both sexually and clonally. We show that this model has a dynamical behaviour that is similar to that of the classical continuous two-dimensional Lotka-Volterra model under weak nonlinearities of the BevertonHolt type. By allowing the species to have different competition efficiencies, we show that it is possible to obtain different dynamics including coexistence, bistability and competitive exclusion, in contrast with the model studied in [2], which exhibits only competitive exclusion behaviour.