Ecological Archives A/E/M000-000-A1
Peter A. Abrams, Lasse Ruokolainen, Brian J. Shuter, and Kevin S. McCann. 2012. Harvesting creates ecological traps: consequences of invisible mortality risks in predator-prey metacommunities. Ecology 93:281–293.
Appendix A. The conditions for extinction and persistence in a model with linear functional responses, the parameter space leading to alternative equilibria in this linear response model, and the responses of populations and yields to harvesting in models with type-2 functional responses.
Conditions for extinction/nonextinction under the linear response model with Eq. 1d
A sufficient condition for extinction to be impossible is that growth within the refuge must be greater than movement out of the refuge when both predator populations are at near-zero density. Given the growth rate under the linear response model with movement according to Eq. 1d, this condition is:
If this condition is satisfied, it is possible to calculate the minimum population of predators in the refuge under extremely high harvest rates in the exploited patch. Under those conditions, the harvested patch will have near-zero predator density and the refuge population will be given by the condition that per capita growth equals per capita movement; i.e., the solution of:
These results also make it possible to calculate a minimum degree of heterogeneity, below which extinction of the refuge (and therefore of the global population) is impossible. Given our assumption that heterogeneous systems have the same mean parameter value this can be done by expressing one of the two patch parameters in Eq. A.1 in terms of the other, setting Eq. A.1 to an equality, and solving (numerically) for that parameter. In the case of the attack rate, we can substitute C2 = 2 - C1 in Eq. A.1, assume the other parameters used in Fig. 2 in the text, and find that the minimum C1 that will allow global extinction at an arbitrarily large harvesting rate in patch 1 is C1 = 1.08021. Similarly the asymptotes of the near vertical lines on the left hand side of Fig. 1 (in the main text) can be determined to be: 1.0119, 1.0347, and 1.0484 for the solid, short-dashed and long-dashed lines in Panel A. There is no heterogeneous system in which the refuge goes extinct at high harvest rates for the solid line under the parameters of Fig. 1B; the asymptotes for the long and short dashed line in Fig. 1B are 1.360, and 1.465 respectively.
Parameter space producing alternative attractors in models with linear response model with Eq. 1d
In the following three graphs (Fig. A1-A3), the mean values of parameters that are identical across the two patches are as in Fig. 2 in the text: dav = 0.025; rav = 1; kav = 1; Cav = 1. We do not consider heterogeneity in d, because it produces results very similar to heterogeneity in r. In each figure, all parameters but one are assumed identical across patches. The single heterogeneous parameter retains its average value. Thus, for each of the cases shown below, the values of the parameter on the x-axis in the two patches sum to 2; thus C2 = 2 - C1 in the first figure. In each figure, the upper line gives the largest harvest rate that produces alternative states, while the lower line is the smallest harvest rate producing alternative states for the corresponding value of the parameter on the x-axis. At least one of the patches has no predators for H values above the upper line. Both patches have predators for parameter values below the lower line. ("No predators" is defined as a population density less than 106.) One of the alternative states involves extinction of at least one predator population. Very low heterogeneities do not have extinction or alternative states. For the values of m and λ used here, the minimum heterogeneity that allows extinction has a patch-1 parameter value of between 1.05 and 1.1. Alternative states do not occur for any harvest rate when heterogeneity in r is larger than the maximum value on the x-axis of Fig. A2.
The x-axis differs in Fig. A3 because the risky (harvested) patch is more attractive at low predator densities when it has a lower k than the refuge, and heterogeneity therefore decreases from left to right on the graph. The maximum and minimum harvest rates yielding alternative states when k1 = 0.9 are 0.365 and 0.240 respectively.
Harvesting and equilibria in models with type-2 responses
Fig. A4. Examination of the responses to harvesting in a model that has type-2 responses but is otherwise comparable to Fig. 2 in the text. It assumes heterogeneity in Ci with C1 + C2 = 2. The other parameters are: ri = ki = 1, b = 1; di = 0.025; and h = 3. This yields the same maximum per capita predator growth rate and the same equilibrium resource density as does the linear functional response model of Fig. 2 when C1 = C2 = 1. Fig. A4 shows how harvesting affects temporal mean predator population sizes and yield when C1 = 1.2 and C2 = 0.8. In panels A and B, the fitness sensitivity is λ = 150, as in Fig. 2; here harvesting cannot cause extinction in the refuge. Panels C and D are identical except that λ = 250; here there is abrupt extinction from high densities at H = 0.401. The global extinction equilibrium is then locally stable down to H = 0.2366. This represents close to an order of magnitude wider range of H values producing alternative occupancy or extinction of both patches than in the comparable linear response system from Fig. 2C. Even higher heterogeneity still produce a wide range of harvest rates with either both patches occupied or extict when λ = 250. If C1 is increased to 1.3, these alternatives occur from H = 0.222 to H = 0.362. Shifts between complex cycles and chaotic dynamics cause the irregular changes in mean predator densities a low harvest rates.
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