ANALYSIS OF MULTI-SPECIES ECOLOGICALAND EVOLUTIONARY DYNAMICS

Ecole Normale Supérieure, ParisDecember 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 readingsEncyclopedia of Theoretical Ecology, Univ. California Press, 2012, pp. 88-95Proc. Roy. Soc. Lond. B (2002) 269:773-780

2.

The Lotka-Volterra competition modelCompetition 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

Competition within two populations

Equilibria and isoclinesequilibria : and

the curves in the state planewhere and

isoclines :

the direction of trajectories :

the principle of competitive exclusion(Hardin G., Science 131, 1960; Gause G.F., Williams&Wilkins, 1934)

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

Four possible scenarios (state portraits)

coexistence dominance-2 dominance-1 mutual exclusion

Back to the principle of competitive exclusion, consider the case ofsymmetric competition with

Mutual exclusion is the resulting scenario when competition is sufficiently strong

A simple model of mutualismTwo 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 andthus is the prob. that an individual of species 2 receives a benefit from species 1 in the time intervalsimilarly for

Equilibria and isoclines

equilibria : and

the direction of trajectories :

The evolution set geometric view: collision and disappearance of two equilibriaalgebraic view : a zero eigenvalue in the system’s Jacobian

The saddle-node bifurcation (see f.r. 1)