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 ecological and evolutionary dynamics

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


Analysis of multi species ecological and evolutionary dynamics

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


Analysis of multi species ecological and evolutionary dynamics

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)


Analysis of multi species ecological and evolutionary dynamics

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


Analysis of multi species ecological and evolutionary dynamics

Four possible scenarios (state portraits)

dominance-2

dominance-1

mutual exclusion

coexistence


Analysis of multi species ecological and evolutionary dynamics

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


Analysis of multi species ecological and evolutionary dynamics

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


Analysis of multi species ecological and evolutionary dynamics

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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