Mutualism
Mutualism is an interaction in which both species receive a net biological benefit. The benefit may increase growth, reproduction or survival, or reduce mortality.
Begin with two self-limited populations
Let \(N_1(t)\) and \(N_2(t)\) be the two population sizes. Without mutualism, suppose each follows logistic growth:
\[\frac{dN_1}{dt}=r_1N_1\left(1-\frac{N_1}{K_1}\right),\qquad \frac{dN_2}{dt}=r_2N_2\left(1-\frac{N_2}{K_2}\right).\]The logistic terms provide intraspecific limitation.
A simple positive interaction
A first model might add positive mass-action terms:
\[\frac{dN_1}{dt}=r_1N_1\left(1-\frac{N_1}{K_1}\right)+\alpha N_1N_2,\]\[\frac{dN_2}{dt}=r_2N_2\left(1-\frac{N_2}{K_2}\right)+\beta N_1N_2.\]Here \(\alpha>0\) and \(\beta>0\) represent beneficial effects. They need not be equal because the two species may depend on each other to different degrees.
Saturating mutualistic benefit
A bounded alternative is to let the per-capita benefit approach a maximum:
\[B_1(N_2)=\frac{a_1N_2}{1+h_1N_2},\qquad B_2(N_1)=\frac{a_2N_1}{1+h_2N_1}.\]As partner abundance becomes large,
\[B_1(N_2)\to\frac{a_1}{h_1},\qquad B_2(N_1)\to\frac{a_2}{h_2}.\]The benefit therefore increases at low partner density but cannot increase indefinitely.
A bounded mutualism model
\[\boxed{\frac{dN_1}{dt}=N_1\left[r_1\left(1-\frac{N_1}{K_1}\right)+\frac{a_1N_2}{1+h_1N_2}\right]},\]\[\boxed{\frac{dN_2}{dt}=N_2\left[r_2\left(1-\frac{N_2}{K_2}\right)+\frac{a_2N_1}{1+h_2N_1}\right]}.\]The self-limiting terms become increasingly negative as each population grows, while the partner benefits remain bounded. This prevents the artificial unlimited positive feedback of the simple linear interaction.
Facultative and obligate mutualism
A facultative mutualist can persist without its partner, although the partner improves its growth or equilibrium abundance. An obligate mutualist has negative growth when the partner is absent and requires enough partner benefit to persist.
For example, if species 1 has baseline per-capita growth \(-d_1\) when rare, then it can increase only when
\[-d_1+\frac{a_1N_2}{1+h_1N_2}>0.\]This can create a minimum partner abundance required for persistence.
Mutualism changes equilibria
At a positive equilibrium, both per-capita growth rates are zero:
\[r_1\left(1-\frac{N_1}{K_1}\right)+\frac{a_1N_2}{1+h_1N_2}=0,\]\[r_2\left(1-\frac{N_2}{K_2}\right)+\frac{a_2N_1}{1+h_2N_1}=0.\]These equations are solved simultaneously. Mutualistic benefit can raise equilibrium abundance above the single-species carrying capacities, but the exact values depend on all parameters.
Mutualism versus other interactions
| Interaction | Effect on species 1 | Effect on species 2 |
|---|---|---|
| competition | negative | negative |
| predation | negative for prey | positive for predator |
| mutualism | positive | positive |
The signs describe the direction of the interaction, but the mathematical form determines how its strength changes with abundance.
Benefits and costs
Mutualistic interactions can include both benefits and costs. A plant may spend energy producing nectar while receiving pollination. A more detailed model may therefore represent the net interaction as benefit minus cost.
An interaction can also change character with density. A partner that is beneficial at moderate abundance may become neutral or harmful if it imposes large costs at high abundance.
Spatial and stochastic effects
Mutualistic partners must often be close enough to interact, so habitat fragmentation can weaken the effective benefit. In small populations, random deaths or failed encounters can also remove one partner and increase extinction risk for the other.
Stochastic mutualism models are therefore useful when persistence probabilities matter more than a single deterministic equilibrium.