Analysis of multi species ecological and evolutionary dynamics
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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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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


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


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)


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)

dominance-2

dominance-1

mutual exclusion

coexistence


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


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


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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