Competition systems with periodic coefficients: A geometric approach

Mottoni, P. de; Schiaffino, A.

doi:10.1007/bf00276900

Cited by 172 publications

(97 citation statements)

References 5 publications

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“…The latter inequality implies instability. It now follows from the results of de Mottoni and Schiaffino [7] that the remaining periodic solution Pa -= P2 -= P > 0 is globally attracting.…”

Section: Discussionmentioning

confidence: 80%

“…In place of the fundamental questions concerning the existence and stability of positive equilibria, one now has the mathematically more difficult questions concerning the existence and stability of positive periodic solutions. Recent general theorems and techniques [4,5,7] can be applied to these fundamental questions or, in the case considered here of small amplitude periodicities, standard small parameter perturbation techniques also yield existence and stability. In the latter case, these regular perturbation techniques can also yield specific results concerning the properties of the solution and its stability as they depend on the properties of the periodicities in the model parameters.…”

Section: Discussionmentioning

confidence: 99%

“…The results and techniques in [4,5,7] could be applied to these existence and stability questions. However, in this paper a different approach based on regular perturbation techniques is taken in order to gain more specific and detailed results on the qualitative properties of the solutions.…”

Section: The Model Equationsmentioning

confidence: 99%

“…Existence and stability questions for more general periodic, two species competition systems were studied in [7] and for general periodic, n-species Kolmogorov systems in [5]. These latter papers deal mainly with the general questions of the mathematical existence and stability of positive periodic solutions, although they do contain some examples which serve to illustrate the fact that periodicities in system parameters can induce unexpected results (such as stable coexistence when averaged coefficients would predict competitive exclusion).…”

Section: Introductionmentioning

confidence: 99%

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Periodic Lotka-Volterra competition equations

Cushing

1986

J. Math. Biology

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Abstract. The Lotka-Volterra competition equations with periodic coefficients derived from the MacArthur-Levins theory of a one-dimensional resource niche are studied when the parameters are allowed to oscillate periodically in time. Specifically, niche positions and widths, resource availability and resource consumption rates are allowed small amplitude periodicities around a specified mean value. Two opposite cases are studied both analytically and numerically. First only resource consumption rates are allowed to oscillate while niche dimensions and resource availability are held constant. The resulting oscillations in population densities and the strength of the system stability as they depend upon crucial relative phase and amplitude differences between the species' consumption rates are studied. This leads to a clear notion of "temporal niche" and of the effects that such oscillations can have on competitive coexistence. Secondly, all system parameters are allowed to oscillate, although the oscillatory consumption rates are assumed identical for both species. The effects on the population density oscillations and their averages are studied and the "best" choice of the common, periodic resource consumption rate for these two "identical" species competing for similar (even identical) niches is considered.

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“…The latter inequality implies instability. It now follows from the results of de Mottoni and Schiaffino [7] that the remaining periodic solution Pa -= P2 -= P > 0 is globally attracting.…”

Section: Discussionmentioning

confidence: 80%

Section: Discussionmentioning

confidence: 99%

Section: The Model Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Periodic Lotka-Volterra competition equations

Cushing

1986

J. Math. Biology

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“…There exists a number of papers which deal with (3.2) for n = 2 with time periodic coefficients, see [5,12,13,36]. We will discuss here the approach of Rosenblat [44], as class of (7. it gives a good impression how asymptotic methods can be applied to this problems.…”

Section: Periodically Forced Volterra-lotka Systemsmentioning

confidence: 99%

Asymptotic methods in mathematical biology

Grasman

1983

Lecture Notes in Mathematics

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Topological properties of strongly monotone planar vector fields

Balanov

Большаков

Рачинский

2019

Mathematische Nachrichten

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We consider strongly monotone continuous planar vector fields with a finite number of fixed points. The fixed points fall into three classes, attractors, repellers and saddles. Naturally, the relative positions of the fixed points must obey a set of restrictions imposed by monotonicity. The study of these restrictions is the main goal of the paper. With any given vector field, we associate a matrix describing the arrangement of the fixed points on the plane. We then use these matrices to formulate simple necessary and sufficient conditions which allow one to determine whether a finite set of attractors, repellers and saddles at given positions on the plane can be realized as the fixed point set of a strongly monotone vector field.

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Competition systems with periodic coefficients: A geometric approach

Cited by 172 publications

References 5 publications

Periodic Lotka-Volterra competition equations

Periodic Lotka-Volterra competition equations

Asymptotic methods in mathematical biology

Topological properties of strongly monotone planar vector fields

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