1 / 8

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 )

sona
Download Presentation

ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related