Stabilizing spatially-structured populations through adaptive Limiter Control

Abstract

Stabilizing the dynamics of complex, non-linear systems is a major concern across several scientific disciplines including ecology and conservation biology. Unfortunately, most methods proposed to reduce the fluctuations in chaotic systems are not applicable to real, biological populations. This is because such methods typically require detailed knowledge of system specific parameters and the ability to manipulate them in real time; conditions often not met by most real populations. Moreover, real populations are often noisy and extinction-prone, which can sometimes render such methods ineffective. Here, we investigate a control strategy, which works by perturbing the population size, and is robust to reasonable amounts of noise and extinction probability. This strategy, called the Adaptive Limiter Control (ALC), has been previously shown to increase constancy and persistence of laboratory populations and metapopulations of Drosophila melanogaster. Here, we present a detailed numerical investigation of the effects of ALC on the fluctuations and persistence of metapopulations. We show that at high migration rates, application of ALC does not require a priori information about the population growth rates. We also show that ALC can stabilize metapopulations even when applied to as low as one-tenth of the total number of subpopulations. Moreover, ALC is effective even when the subpopulations have high extinction rates: conditions under which another control algorithm had previously failed to attain stability. Importantly, ALC not only reduces the fluctuation in metapopulation sizes, but also the global extinction probability. Finally, the method is robust to moderate levels of noise in the dynamics and the carrying capacity of the environment. These results, coupled with our earlier empirical findings, establish ALC to be a strong candidate for stabilizing real biological metapopulations.

Figure 1. Effects of ALC on metapopulation FI and synchrony at different rates of migration.

(A). Both LALC (c = 0.25) and HALC (c = 0.4) increases metapopulation FI at low migration rates, but reduces the same at high migration rates. This contrasting effect can be explained by (B) which shows that ALC reduces both positive and negative synchrony, which in turn is expected to have opposite effects on metapopulation constancy. N₀ = 20 and K = 30 for all figures including this one. Intrinsic growth rate, r = 3.5 (for this and all subsequent figures except Figure 2). Each point in every figure is a mean of 100 independent runs. Error bars denote ±SEM and are too small to be visible.

Figure 2. Effects of ALC on metapopulation stability at different intrinsic growth rate (r) values.

ALC enhances (A) constancy and (B) persistence over a wide parameter range, and has no effects in other zones. See main text for a possible explanation. Migration rate (m) = 0.3 in this and all subsequent figures. Error bars denote ±SEM and are too small to be visible.

Figure 3. Effects of ALC on metapopulation constancy under different rates of subpopulation extinction.

(A). With increasing extinction probability when the population size goes below 4. (B) With increasing critical population sizes below which, there was a 50% extinction probability that the population would go extinct. In both cases, increasing the rate of extinction did not reduce the efficacy of ALC in inducing greater constancy. See main text for a possible explanation. Error bars denote ±SEM and are too small to be visible.

Figure 4. Effects of ALC on constancy in metapopulations with different number of subpopulations.

(A) LALC (i.e. c = 0.25), and (B) HALC (i.e. c = 0.4). In both figures, only one subpopulation is perturbed for increasing number of subpopulations. Perturbing only 1 patch by ALC can reduce FI of metapopulations with up to 10 subpopulations. Error bars denote ±SEM and are too small to be visible.

Figure 5. Effects of increasing the fraction of ALC controlled subpopulation on metapopulation constancy.

In this figure, each metapopulation consists of 10 subpopulations. For low values of c, increasing the fraction of perturbed subpopulations can have a negative effect on constancy. Error bars denote ±SEM and are too small to be visible.