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ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS

ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS. 2. Simple models of competition and mutualism (F. Dercole )

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ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS

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  1. ANALYSIS OF MULTI-SPECIES ECOLOGICALAND EVOLUTIONARY DYNAMICS 2. 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 Ecole Normale Supérieure, Paris December 9-13, 2013

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

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

  4. 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 system’s Jacobian

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

  6. 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

  7. A simple model of mutualism Two species, e.g. flowers and pollinating insects, with densities and 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 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

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

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