Multiple resource limitation: nonequilibrium coexistence of species in a competition model using a synthesizing unit

Abstract

During the last two decades, the simple view of resource limitation by a single resource has been changed due to the realization that co-limitation by multiple resources is often an important determinant of species growth. Hence, the multiple resource limitation hypothesis needs to be taken into account, when communities of species competing for resources are considered. We present a multiple species–multiple resource competition model which is based on the concept of synthesizing unit to formulate the growth rates of species competing for interactive essential resources. Using this model, we demonstrate that a more mechanistic explanation of interactive effects of co-limitation may lead to the known complex dynamics including nonequilibrium states as oscillations and chaos. We compare our findings with earlier investigations on biological mechanisms that can predict the outcome of multispecies competition. Moreover, we show that this model yields a periodic state where more species than limiting complementary resources can coexist (supersaturation) in a homogeneous environment. We identify two novel mechanisms, how such a state can emerge: a transcritical bifurcation of a limit cycle and a transition from a heteroclinic cycle. Furthermore, we demonstrate the robustness of the phenomenon of supersaturation when the environmental conditions are varied.

Appendices

Appendix 1: SU rule for k resources

In a homogeneous environment, the arrival fluxes J _j are proportional to the concentrations R _j of resources X _j and n _j J _m are proportional to the half-saturation constants K _ji for species Y _i . Scaling J _m = 1 as in Eq. 1, the SU rule for multiple (say k) limiting (interactive essential or complementary) resources for species Y _i can be written as follows (Muller et al. 2001; Kooijman et al. 2004; Kooijman 2010):

$${\small\begin{aligned} f_{ji} = &\;\left( 1+\sum\limits_{j_{1}=1}^{k}\left(\frac{R_{{j_{1}}}}{K_{j_{1}i}}\right)^{-1} -\sum\limits_{j_{2} > j_{1}=1}^{k}\left(\sum\limits_{q=1}^{2}\frac{R_{j_{q}}}{K_{j_{q}i}}\right)^{-1}\right.\\ &\left.+\sum\limits_{j_{3} > j_{2} > j_{1}=1}^{k}\left(\sum\limits_{q=1}^{3}\frac{R_{j_{q}}}{K_{j_{q}i}}\right)^{-1}- {\dots} \right.\\ & \left.{\dots} -(-1)^{k}\sum\limits_{j_{k} > {\dots} >j_{1}=1}^{k}\left(\sum\limits_{q=1}^{k}\frac{R_{j_{q}}}{K_{j_{q}i}}\right)^{-1}\right)^{-1}. \end{aligned}}$$

Appendix 2: Parameter values used in our simulations

Table 2 Three species competing for three resources: D = 0.25, S ₁ = S ₂ = S ₃ = 20, m ₁ = 0.015, m ₂ = 0.01, and m ₃ = 0.018

Table 3 Four species competing for three resources: D = 0.22, S ₁ = S ₂ = S ₃ = 35, m ₁ = 0.015, m ₂ = 0.01, m ₃ = 0.0175, and m ₄ = 0.016

Table 4 Five species competing for three resources: D = 0.21, S ₁ = S ₂ = S ₃ = 25, m ₁ = 0.013, m ₂ = 0.01, m ₃ = 0.014, m ₄ = 0.0125, and m ₅ = 0.018

Table 5 Six species competing for three resources: D = 0.23, S ₁ = S ₂ = S ₃ = 32, m ₁ = 0.013, m ₂ = 0.01, m ₃ = 0.014, m ₄ = 0.0125, m ₅ = 0.018, and m ₆ = 0.016

Online Appendix–III: Time series for different parameter values of three species competing for three resources

Fig. 10

Competition between three species (labeled as 1, 2 and 3 in (a) for three essential resources. (a) Competitive exclusion where the winner is depending upon the initial conditions, (b) species oscillations leading to a heteroclinic cycle, (c) species oscillations leading to a limit cycle, and (d) stable equilibrium coexistence of all the competing species. The parameter values are given in Table 2 in Appendix 2