## Mutualism

Mutualism has been somewhat neglected by ecologists, compared to competition, but stoichiometric approaches again apply, at least for mutualisms based on production of substances beneficial to growth of another population. Consider two populations in a symmetrical mutualism where each population consumes a resource produced by the other population: species 1 consumes resource 1 and produces resource 2, while species 2 consumes resource 2 and produces resource 1. Assume also that the two resources are also supplied by other processes.

The graphical display of this interaction follows a now familiar path: plot ZNGI graphs, mass-conservation constraints, and impact vectors. For each population, its ZNGI graph is a line parallel to the axis for the resource it produces but does not require (Figure 7), an

Nutrient 1 concentration, R1

Figure 7 A symmetric mutualism involving two populations. ZNGI graphs (indicated by Zi) intersect at the circled point, making steady-state coexistence possible. The shaded feasible region for steady states is superimposed, bounded by mass-conservation constraints for populations 1 and 2 (indicated by M), as are impact vectors implying stable coexistence (indicated by I).

Nutrient 1 concentration, R1

Figure 7 A symmetric mutualism involving two populations. ZNGI graphs (indicated by Zi) intersect at the circled point, making steady-state coexistence possible. The shaded feasible region for steady states is superimposed, bounded by mass-conservation constraints for populations 1 and 2 (indicated by M), as are impact vectors implying stable coexistence (indicated by I).

arrangement that is guaranteed to make coexistence possible. For each population, its mass-conservation constraint expresses the stoichiometry of the resource it produces relative to the resource it consumes, and steady-state coexistence is feasible when these bounds enclose the intersection of the ZNGI graphs. For each population, its impact vector expresses the same stoichiometry of production and consumption, and stability in this example is guaranteed because each population has a negative impact on its own growth through consumption of its own required resource.

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