# Mutualism (biology)

Mutualism or mutualistic symbiosis describes in ecology an interrelation between living beings of two species , from which both partners benefit in contrast to competition or predator-prey relationship . In US-American literature, the word mutualism is used synonymously with the German term symbiosis in the narrow sense, while the word symbiosis - in contrast to the German term symbiosis - is used for any coexistence of living beings of different species. Some authors differentiate between a symbiotic mutualism, in which both partners live together spatially, and a non-symbiotic mutualism, (e.g. in zoogamy or zoochory ), in which the symbiotic partners only meet sporadically.

Various mathematical models are used in biology to describe and analyze mutualistically organized ecological relationships. Such models differ, among other things, according to whether they describe the temporal dynamics of mutually interacting populations in a rather short (ecological) or a rather long (evolutionary) period. A particularly simple model, popular because of its broad applicability, can be written for mutualisms of two species in the form of ordinary differential equations : N 1 and N 2 denote the population densities of two species in a mutualistic relationship, r 1 and r 2 intrinsic growth constants, K 1 and K 2 capacities as well as b 12 and b 21 positive interaction coefficients, which indicate the strength of the mutualistic relationship, one obtains:

${\ displaystyle {\ frac {\ mathrm {d} N_ {1}} {\ mathrm {d} t}} = r_ {1} N_ {1} \ left [1 - {\ frac {N_ {1}} { K_ {1}}} + b_ {12} {\ frac {N_ {2}} {K_ {1}}} \ right]}$

${\ displaystyle {\ frac {\ mathrm {d} N_ {2}} {\ mathrm {d} t}} = r_ {2} N_ {2} \ left [1 - {\ frac {N_ {2}} { K_ {2}}} + b_ {21} {\ frac {N_ {1}} {K_ {2}}} \ right]}$

This type of model is a direct generalization of the logistic differential equation to the situation of two interacting populations. Mutualisms are given in this or a similar form in many textbooks on theoretical biology (see literature).