The struggle for life: I. Two species

Rescigno, Aldo; Richardson, I.W.

doi:10.1007/bf02476909

Cited by 100 publications

(28 citation statements)

References 7 publications

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“…For further discussion of the use of cooperative or competitive systems as biological models the reader may consult, among many other works, A. Rescigno and I. Richardson (1967), A. N. Kolmogorov (1936), S. Grossberg (1978, S. Smale (1976), W. Leonard andR. May (1975), P. Waltman (1983), H. Freedman (1980).…”

Section: Tof(y)mentioning

confidence: 99%

The dynamical systems approach to differential equations

Hirsch¹

1984

Bull. Amer. Math. Soc.

289

156

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Section: Tof(y)mentioning

confidence: 99%

The dynamical systems approach to differential equations

Hirsch¹

1984

Bull. Amer. Math. Soc.

289

156

View full text Add to dashboard Cite

“…The functions f and g describe the nature of the interactions between populations. Models in this form have a long history of investigation, for example, having been studied by Kolmogoroff, [20], Rescigno and Richardson, [26], [27], May, [23], and Albrecht, Gatzke, and Haddad and Wax, [1]. See Chapter 5 of Freedman, [6], for a thorough discussion of such models.…”

Section: Predator-prey Systemsmentioning

confidence: 99%

Did Something Change? Thresholds in Population Models

Hoppensteadt¹,

Waltman

2003

Trends in Nonlinear Analysis

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Dedication: This paper is dedicated to our good friend Professor Willi Jäger on the occasion of his 60th birthday. The three of us have collaborated since a momentous year (for us) at the Courant Institute of Mathematical Sciences in 1969-70.Abstract: The goal of this article is to illustrate several interesting bifurcations that can arise in population biology. These are of interest since it is often through bifurcation phenomena that changes significant enough to be measured occur. For example, a minor change in some environmental parameter can cause a system to change from being at rest to oscillating. We illustrate here the role of several canonical types of bifurcations in population modeling. IntroductionThe subject of mathematical population biology is a vast one, much too large to be surveyed in a single article. Since the thrust of this volume is to present material in a way to make the subject attractive to students, we have chosen to focus on specific examples and to illustrate the phenomenon of change. This forms one of the connections between theories and experiments since experiments often focus on when a moderate change in a parameter produces a major change in the outcome. Such changes in the model's solutions can reflect the occurrence of a bifurcation of solutions and with it an exchange of stabilities. For example, a stable static state can bifurcate into a new stable state or a stable oscillation as model parameters change. We present here several examples of where bifurcation phenomena have been observed in population biology and how they have been used to explain, predict and control outcomes. A major proponent of this approach is Rene Thom [30] who used canonical forms of singularities to identify bifurcation phenomena in developmental biology. In [14] a systematic development of certain bifurcations that arise in neuroscience is presented, and there are many excellent texts that present bifurcation phenomena for mathematicians and engineers, for example [2, 8,

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“…Various outcomes are possible inside this curve including nested limit cycles, neutrally stable closed trajectories, and asymptotic approaches to the equilibrium point, the latter being either stationary (node) or oscillatory (f ocus) in nature Richardson, 1967, 1973). Neither Kolmogorov ( 1936 ) nor Rescigno and Richardson ( 1967 ) were able to find a necessary and sufficient condition for the existence of periodic limit cycle solutions, although they concluded that the community equilibrium point being linearly unstable was a sufficient condition for the existence of the closed integral curve (Rescigno and Richardson, 1973). As an initial check on the validity of May's assertion, we apply a numerical differential-equation solver to our composite Holling-Tanner model for various relevant temperatures in the specific intervals delineated earlier, and for a number of different possible combinations of initial conditions.…”

Section: O=o(t)p=dp(t)=¢o~(t) For0mentioning

confidence: 99%

Temperature-mediated stability of the interaction between spider mites and predatory mites in orchards

1988

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The nonlinear behavior of the Holling-Tanner predatory-prey differential equation system, employed by R.M. May to illustrate the apparent robustness of Kolmogorov's Theorem when applied to such exploitation systems, is re-examined by means of the numerical bifurcation code AUTO 86 with model parameters chosen appropriately for a temperature-dependent mite interaction on fruit trees. The most significant result of this analysis is that there exists a temperature range wherein multiple stable states can occur, in direct violation of May's interpretation of this system's satisfaction of Kolmogorov's Theorem: namely, that linear stability predictions have global consequences. In particular these stable states consist of a focus (spiral point) and a limit cycle separated from each other in the phase plane by an unstable limit cycle, all of which are associated with the single community equilibrium point of the system. The ecological implications of such metastability, hysteresis, and threshold behavior for the occurrence of outbreaks, the persistence of oscillations, the resiliency of the system, and the biological control of mite populations are discussed.

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The struggle for life: I. Two species

Cited by 100 publications

References 7 publications

The dynamical systems approach to differential equations

The dynamical systems approach to differential equations

Did Something Change? Thresholds in Population Models

Temperature-mediated stability of the interaction between spider mites and predatory mites in orchards

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