Stability of discrete age-structured and aggregated delay-difference population models

Bergh, M. O.; Getz, Wayne M.

doi:10.1007/bf00276060

Cited by 54 publications

(24 citation statements)

References 23 publications

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“…If no individuals survive beyond age n, then this model takes that form (Bergh & Getz 1988;Getz & Haight 1989;Kaitala & Getz 1995) ( 7.27) where y t is the number of young produced at time t and (7.28) is the 'breeding density' of the population at time t (see Gatto 1993 for a simpler approach to modelling iteroparous fish stocks). The parameters c j are themselves constructed from fecundity and survivorship parameters, b i and s i respectively, where b i is the average number of females born to a female of age i, s i is the probability of surviving from age i to age i + 1, and hence ഞ i = '…”

Section: Age-structured Populationsmentioning

confidence: 99%

Population and Evolutionary Dynamics of Consumer‐Resource Systems

Getz

1999

Advanced Ecological Theory

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IntroductionConsumer-resource systems can be regarded as any system in which a population of organisms exploits either an energy flux, a detritus or nutrient pool, or another population. This definition then includes plants growing in sunlight, parasites growing in or on hosts, predators exploiting prey, herbivores grazing or browsing on plants, and even humans harvesting trees, fish or game. For such a vast array of systems, no single best paradigm exists for modelling the dynamics of the populations under consideration.Here I will focus on two paradigms: a discrete-time paradigm that is suitable for modelling populations with nonoverlapping generations and a continuous-time paradigm that is suitable for modelling the flow of biomass from the resource to the consumer, and its conversion from resource mass (biomass, nutrients, or energy) into consumer mass. I will also consider the extension of the discrete-time paradigm to include overlapping generations (age structure) and the application of the models to resource management problems.The discrete-time models fall within the class of iterative maps (7.1)where, for the systems considered here, the elements of the vector x t = (x 1t , x 2t , . . ., x nt )¢ (¢denotes the transpose of a vector) typically represent the densities of individuals in n biological populations at time t = 0, 1, 2, 3, . . . or the densities of individuals in n classes (e.g., age or size) of one population at time t, or even some combination of several populations each with several classes of individuals. The continuous-time models fall within the class of vector differential equation systemswhere, for the systems considered here, the elements of the vector x(t) = (x 1 (t), x 2 (t), . . ., x n (t))¢ typically represent the biomass densities of n interacting biological populations at time d d x x t = ( ) y , x x t t t + = ( ) = 1 0 1 2 y , , , . . .

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Section: Age-structured Populationsmentioning

confidence: 99%

Population and Evolutionary Dynamics of Consumer‐Resource Systems

Getz

1999

Advanced Ecological Theory

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“…Chaotic behaviour has earlier been proposed by May (1976) for laboratory and field populations of insects, by Powers (1989) for a two species system of fish and by Schaffer and Kot (1986) and Kot et al (1988) for outbreaks of insects pests and of human diseases. Also, Berg and Getz (1988) suggested that stock and recruitment, in a sardine-like population, moved along a path or attractor in some higher dimension coordinate system and Conan (1994) observed that lobster and snow crab landings in Atlantic Canada may follow two orbits of stability or cycles.…”

Section: Multiple Equilibria and Chaos To Cyclesmentioning

confidence: 99%

Stock and recruitment in Baltic cod (Gadus morhua): a new, non-linear approach

Solari¹

1997

ICES Journal of Marine Science

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The stock and recruitment (SR) relationship in Baltic cod is studied. In light of the field data, the authors suggest that the SR relationship oscillates at two levels of stability (cycles) due to the interaction between density-dependent compensation and depensation. Furthermore, density-independent transitions between equilibrium states are assumed due to fishing mortality and medium and long-term periodicities in the abiotic environment which may induce compensatory and depensatory effects upon stock and recruitment. These mechanics are related to concepts such as variable carrying capacity, multiple equilibria, minimum viable population and inverse density dependence. Carrying capacity is regarded as a critical threshold between different equilibrium states and the minimum viable population as an unstable equilibrium below which the SR relationship may not rehabilitate. A modelling approach is put forward where the SR-relationship is regarded as a system or summation of non-linear functions with dynamic features ranging from chaos (the ceiling, when external conditions are extremely benign), going through a range of relatively stable, converging cycles (as external stress increases), to a quasi-standstill state with no clear oscillations (when the minimum viable population is being approached) which may lead to inverse density-dependence (depensatory dynamics). This SR-system is considered as highly flexible as it has the capacity to, persistently, evolve and return within a range of equilibrium states. Also, it is proposed that the SR relationship is, at the present time, nearby the minimum viable population due to the combined effects from high fishing mortality and negative effects from external perturbations. However, a SR rehabilitation towards a low equilibrium state is expected, during the coming years likely due to positive trends in external perturbations. A simple numerical simulation is put forward where the SR system is perturbed by a sinusoidal external variable at three constant levels of mortality.1997 International Council for the Exploration of the Sea

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“…In recent years, stage structure models have received a lot of attention (see, e.g., [8][9][10][11][12][13][14][15][16]). Tang and Chen [17] proposed that a singlespecies population has stage structure, and the population is divided into immature and mature classes, and only the mature population can reproduce as follows:…”

Section: Introductionmentioning

confidence: 99%

Stability and bifurcation analysis for a single-species discrete model with stage structure

Zhao

Zhang

2018

Adv Differ Equ

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In this paper, a single-species discrete model with stage structure is investigated. By analyzing the corresponding characteristic equations, the local asymptotic stability of non-negative equilibrium points and the existence of flip bifurcation are discussed. Using the center manifold theory, the stability of the non-hyperbolic equilibrium point is obtained. Based on bifurcation theory, we obtain the direction and the stability of a flip bifurcation at the positive equilibrium with the birth rate as the bifurcation parameter. Finally, some numerical simulations, including phase portraits, chaotic bands with period windows, and Lyapunov exponent methods, are performed to validate the theoretical results, which extends the results in previous papers.

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Stability of discrete age-structured and aggregated delay-difference population models

Cited by 54 publications

References 23 publications

Population and Evolutionary Dynamics of Consumer‐Resource Systems

Population and Evolutionary Dynamics of Consumer‐Resource Systems

Stock and recruitment in Baltic cod (Gadus morhua): a new, non-linear approach

Stability and bifurcation analysis for a single-species discrete model with stage structure

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