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ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS Ecole Normale Supérieure, Paris December 9-13, 2013 Simple models of competition and mutualism (F. Dercole) The Lotka-Volterra competition model. Symmetric vs asymmetric competition. Equilibria and isoclines. The principle of competitive exclusion. Transcritical bifurcations. A simple model of mutualism. Obligate vs non-obligate mutualism. Equilibria and isoclines. Saddle-node bifurcation. Further readings Encyclopedia of Theoretical Ecology, Univ. California Press, 2012, pp. 88-95 Proc. Roy. Soc. Lond. B (2002) 269:773-780 2.2.

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The Lotka-Volterra competition model Competition within one population (the logistic model) Competition within two populations is the carrying capacity is the intrinsic (or initial) per-capita growth rate is the per-capita competition mortality (adimensional) competition coefficients symmetric competition asymmetric competition favoring population 2 / 1

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Competition within two populations Equilibria and isoclines equilibria :and the curves in the state plane where and isoclines : the direction of trajectories : the principle of competitive exclusion (Hardin G., Science 131, 1960; Gause G.F., Williams&Wilkins, 1934)

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Transcritical bifurcations (see f.r. 1) geometric view: collision of two equilibria, as a parameter is varied, which exchange stability algebraic view: a zero eigenvalue in the systems Jacobian

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Four possible scenarios (state portraits) coexistence dominance-2dominance-1mutual exclusion

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Back to the principle of competitive exclusion, consider the case of symmetric competition with Mutual exclusion is the resulting scenario when competition is sufficiently strong

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A simple model of mutualism Two species, e.g. flowers and pollinating insects, with densities and There is intra-specific competition for commodities, as well as for other resources The mutualism is obligate A simple model (see f.r. 2) where and are nonnegative increasing functions and,,,, are positive constant parameters The per-capita rates of commodities trading are inheritable phenotypes and thus is the prob. that an individual of species 2 receives a benefit from species 1 in the time interval similarly for

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Equilibria and isoclines equilibria :and the direction of trajectories : The evolution set geometric view: collision and disappearance of two equilibria algebraic view: a zero eigenvalue in the systems Jacobian The saddle-node bifurcation (see f.r. 1)

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