The Struggle for Existence

Gause, G. F.

doi:10.1097/00010694-193602000-00018

Cited by 349 publications

(331 citation statements)

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“…Another noteworthy prediction from model (1.1) is that prey and predator species can not extinct simultaneously (mutual extinction). This, however, clearly contradicts Gause's classic observation of mutual extinction in the protozoans, Paramecium and its predator Didinium (Gause (1934), Abrams and Ginzburg (2000)). …”

Section: Introductioncontrasting

confidence: 60%

Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system

Hsu

Hwang

Kuang³

2001

Journal of Mathematical Biology

330

171

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Abstract. The recent broad interest on ratio-dependent based predator functional response calls for detailed qualitative study on ratio-dependent predator-prey differential systems. A first such attempt is documented in the recent work of Kuang and Beretta(1998), where Michaelis-Menten-type ratio-dependent model is studied systematically. Their paper, while contains many new and significant results, is far from complete in answering the many subtle mathematical questions on the global qualitative behavior of solutions of the model. Indeed, many of such important open questions are mentioned in the discussion section of their paper.Through a simple change of variable, we transform the Michaelis-Menten-type ratiodependent model to a better studied Gause-type predator-prey system. As a result, we can obtain a complete classification of the asymptotic behavior of the solutions of the MichaelisMenten-type ratio-dependent model. In some cases we can determine how the outcomes depend on the initial conditions. In particular, open questions on the global stability of all equilibria in various cases and the uniqueness of limit cycles are resolved. Biological implications of our results are also presented.

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Section: Introductioncontrasting

confidence: 60%

Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system

Hsu

Hwang

Kuang³

2001

Journal of Mathematical Biology

330

171

View full text Add to dashboard Cite

show abstract

“…This, however, contradicts Gause's classic observation of mutual extinction in the protozoans, Paramecium and its predator Didinium (Gause(1934), Luckinbill(1973), Abrams and Ginzburg(2000)), and the well cited experimental observation of Huffaker (Huffaker(1958)). Until very recently, both ecologists and mathematicians chose to ignore the rich dynamics provided by the strict ratio-dependent models, especially that on the boundary and close to the origin (the origin is a singular equilibrium, which renders direct local stability analysis impossible).…”

Section: Introductionmentioning

confidence: 69%

Rich dynamics of a ratio-dependent one-prey two-predators model

Hsu

Hwang

Kuang

2001

Journal of Mathematical Biology

112

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Abstract. The objective of this paper is to systematically study the qualitative properties of a ratio-dependent one-prey two-predator model. We show that the dynamics outcome of the interactions are very sensitive to parameter values and initial data. Specifically, we show the interactions can lead to all the following possible outcomes: 1) competitive exclusion; 2) total extinction, i.e., collapse of the whole system; 3) coexistence in the form of positive steady state; 4) coexistence in the form of oscillatory solutions; and 5) introducing a friendly and better competitor can save a otherwise doomed prey species. These results reveal far richer dynamics compared to similar prey dependent models. Biological implications of these results are discussed.

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“…The principle of competitive exclusion (Gause 1934, Hardin 1960 states that only the most successful species can survive in a given niche. That is, coexistence requires ecological (niche) segregation; limiting similarity of the coexisting species is expected.…”

Section: Introductionmentioning

confidence: 99%

Coexistence in a fluctuating environment by the effect of relative nonlinearity: A minimal model

Szilágyi

Meszéna

2010

Journal of Theoretical Biology

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The minimal model of the "relative nonlinearity" type fluctuation-maintained coexistence is investigated. The competing populations are affected by an environmental white noise. With quadratic density dependence, the long-term growth rates of the populations are determined by the average and the variance of the (fluctuating) total density. At most two species can coexist on these two "regulating" variables; competitive exclusion would ensue in a constant environment. A numerical study of the expected time until extinction of any of the two species reveals that the criterion of mutual invasibility predicts the parameter-range of long-term coexistence correctly in the limit of zero extinction threshold. However, any extinction threshold consistent with a realistic population size will allow only short-term coexistence. Therefore, our simulations question the biological relevance of mutual invasibility, as a sufficient condition of coexistence, for large density fluctuations. We calculate the average and the variance of the fluctuating density of the coexisting populations analytically via the moment-closure approximation; the results are reasonably close to the simulated behavior. Based on this treatment, robustness of coexistence is studied in the limit of infinite population size. We interpret the results of this analysis in the context of necessity of niche segregation with respect to the regulating variables using a framework theory published earlier.

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The Struggle for Existence

Cited by 349 publications

References 0 publications

Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system

Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system

Rich dynamics of a ratio-dependent one-prey two-predators model

Coexistence in a fluctuating environment by the effect of relative nonlinearity: A minimal model

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