A stochastic bivariate ecology model for competing species

Tsokos, Chris P.; Hinkley, Sidney W.

doi:10.1016/0025-5564(73)90030-8

Cited by 11 publications

(5 citation statements)

References 6 publications

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“…A description of the population requires an infinite number of equations of this form, plus a boundary condition that n = 0 is an absorbing state. A number of studies have extended the birth-death model formulation to more than one interacting species (e.g., Chiang 1954, Bartlett 1957, Leslie and Gower 1958, 1960, Tsokos and Hinckley 1973. A recent application of a birth-anddeath model to an insect population was made by Des-Ecological Monographs charnais and Costantino (1982).…”

Section: From Stable Equilibrium To Stochastic Dominationmentioning

confidence: 99%

“…A recent application of a birth-anddeath model to an insect population was made by Des-Ecological Monographs charnais and Costantino (1982). A number of studies have extended the birth-death model formulation to more than one interacting species (e.g., Chiang 1954, Bartlett 1957, Leslie and Gower 1958, 1960, Tsokos and Hinckley 1973.…”

Section: From Stable Equilibrium To Stochastic Dominationmentioning

confidence: 99%

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Equilibrium and Nonequilibrium Concepts in Ecological Models

Jiang¹,

Waterhouse

1987

Ecological Monographs

518

254

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Mathematical models and empirical studies have revealed two potentially disruptive influences on ecosystems; (1) instabilities caused by nonlinear feedbacks and time-lags in the interactions ofbiological species, and (2) stochastic forcings by a fluctuating environment. Because both of these phenomena can severely affect system survival, ecologists are confronted with the question of why complex ecosystems do, in fact, exist. Our study analyzes the basic themes of this research and identifies five general hypotheses that, in recent years, theoretical ecologists have built into models to increase their stability against disruptive feedback and stochasticity. To counter feedback instabilities, theoreticians have considered (1) functional interactions between species that act as stabilizers, (2) disturbance patterns that interrupt adverse feedback effects, and (3) the stabilizing effect of integrating small-spatial-scale systems into large landscapes. To decrease the influence of stochasticity, modelers have hypothesized (4) compensatory mechanisms operating at low population densities, and (5) the moderating effect of spatial extent and heterogeneity. We show that modeling based on these ideas can be organized in a systematic way. We also show that the stable equilibrium state should not be viewed as a fundamental property of ecological systems, but as a property that can emerge asymptotically from extrapolation to sufficiently large spatial scales.

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Section: From Stable Equilibrium To Stochastic Dominationmentioning

confidence: 99%

Section: From Stable Equilibrium To Stochastic Dominationmentioning

confidence: 99%

Equilibrium and Nonequilibrium Concepts in Ecological Models

Jiang¹,

Waterhouse

1987

Ecological Monographs

518

254

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“…Models similar to this have been considered by Bartlett et al (1960), Barnett (1962), Tsokos and Hinkley (1973), Smith and Mead (1980), Hitchcock (1986) and others. A model of this type could be applied to prey-predator interaction such as: (I) the citrus pest, Icerya purchasi, which is a prey to the predator Rodolia cardinalis, a coccinellid beetle, (2) the noxious weed, Opuntia, a prey to the moth, Cactoblastis cactorum, (3) plytoplankton-zooplankton systems, and perhaps (4) the lynx-snowshoe hare system (Williamson 1972;Varley et al 1974).…”

Section: Inference For a Two Species Modelmentioning

confidence: 99%

Inference on system parameters in a model for interacting species

Sridhara

1990

J. Math. Biol.

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“…The equilibrium points for these models were determined and trajectories were constructed showing the behavior of the system near them. Hinkley and Tsokos 12 have considered the stochastic form of these models and have attempted to examine their stochastic behavior by determining the manner in which the conditional probabilities of the system vary with time. Unfortunately, a term was left out in the final form of their model which made the results not very relevant.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Ecology Models for Two Interacting Populations

Smith

Tsokos

1978

Kybernetes

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The interaction of two species can be considered to be either mutually competitive, the "co-predator" case, or dominated by one species, the "predator-prey" model. The study of stochastic versions of these models has been impeded by the occurrence of nonlinear expressions in the characterization of the resulting probability structure. Generating function methods are used to study the bivariate and conditional probability distributions and their moments. The results are illustrated using a Laplace transform technique for obtaining numerical values. These methods can be used in the development of ecological prediction techniques and can be extended easily to the bivariate logistic model.

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A stochastic bivariate ecology model for competing species

Cited by 11 publications

References 6 publications

Equilibrium and Nonequilibrium Concepts in Ecological Models

Equilibrium and Nonequilibrium Concepts in Ecological Models

Inference on system parameters in a model for interacting species

Stochastic Ecology Models for Two Interacting Populations

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