Model Description
This model presents an example of a global bifurcation (a heteroclinic connection). The model is a three-dimensional system with two resources and a single consumer, where one of the resources is fixed and the other is reproductive. By assuming that, for all values of resource consumers (C) below its carrying capacity (K), the fixed resource facilitates the consumption of the reproductive resource, the system can be reduced to a two-dimensional system. The reference article, Vandermeer and King (2010), gives the following abstract of the model. "Numerous situations exist in which a consumer uses two different kinds of resources, one fixed, the other renewable, e.g., nesting resources and food resources. With an elementary modification of the basic Lotka–Volterra consumer resource equations, we investigate the population dynamics of a consumer dependent on two resources, one fixed, the other renewable. Emerging from this structure is a situation of alternative attractors that remain qualitatively robust over a significant range of parameter values. However, a dramatic change in basins of attraction is induced by very small changes in parameters due to a global bifurcation. Noteworthy is the fact that the qualitative nature of the alternative equilibria remains constant but the dramatic change in the basins does not arise from subtle differences in initial conditions. Rather, there is a major restructuring of the vector field such that a permanent change involving large sets of initial conditions results from very small changes in parameters".
Reference
Vandermeer, J., & King, A. (2010). Consequential classes of resources: Subtle global bifurcation with dramatic ecological consequences in a simple population model. Journal of theoretical biology, 263(2), 237–41. doi:10.1016/j.jtbi.2009.12.006
Scenarios
This model demonstrates that dramatic changes in system behavior can still happen even without so-called regime shifts or critical transitions. To see this, imagine that some invasive pests (C) that devour crops (R) are naturally controlled every year by the presence of their natural enemies (e.g., birds). Let us assume that this happens when the parameters have the following values: a=0.2, K=2.5, m=0.01, g=0.12, and q=1. Under this setting, small introductions of pests will be eventually controlled (C will oscillate toward a stable fixed point) no matter how abundantly available their food (crops) is. Now, imagine that social systems adopt a more advanced technology for cultivating their coprs and somehow this new action increases the crop regeneration rate (e.g., g=0.135). This slight change can create a dramatic change in the population of the invasive pest species. To see this, slightly increase the crop regeneration rate (g=0.135). This is an example of a global bifurcation (a heteroclinic connection).
$\Large \frac{dR}{dt}=gR\left (1-R\right )-qa\left (\frac{K-C}{K} \right )CR$ |
| Rate of change in the renewable resource. This relationship holds only for C < K. |
$\Large \frac{dC}{dt}=a\left (\frac{K-C}{K} \right )CR-mC$ |
| Rate of change in the resource consumers. This relationship holds only for C < K. |
#=====define parameters par a=0.2, K=2.5, m=0.01, g=0.12, q=1 #=====define some hidden variables===== #============economic functions #==================demographicfunctions #=====auxiliary quantities================= #======right hand sides dC/dt=a*((K-C)/K)*C*R-m*C dR/dt=g*R*(1-R)-q*a*((K-C)/K)*C*R #=============initial data init C=0.2, R=0.5 @ meth=qualrk @ bounds=500 @ total=250 @ dt=0.1 @ yp=C, xp=R @ ylo=0,yhi=3, xlo=0, xhi=1.5 done
Yu JHD, Arizona State University.
Bozicevic M, Arizona State University.
