Figure 1.
Design of Computational Model (IBM) Experiments
(A) Schematic diagram of invasion experiments. Initially, a resident species populates the habitat with a finite initial density and is given a finite time period (dashed vertical line) in isolation sufficiently long to evolve an adaptive behavioral strategy. Thereafter individuals from the “invader” species are added to the habitat. In some cases, the resident species will repel invasion, with the invader species density never increasing. In others, as exemplified in the figure, the invader become established and displaces the resident.
(B) Invasion experiments as in (A) were performed for all four combinations of residents and invaders as either NPP-consumers (cannot evolve plasticity) or PP-consumers (can evolve plasticity). Note the resident species evolves an adaptive strategy whether it is an NPP- or PP-resident species, the difference being that, unlike in the PP case, in the NPP case, the resident is constrained to find the behavior that optimizes fitness when that behavior must be the same in both environmental states (predator present and absent).
Figure 2.
Effect of Plasticity on Invasion Time
Mean invasion time (number of steps, log scale) as a function of the background death probability of the resident species, for cases in which the resident and invader both did (PP, solid line) or did not (NPP, empty line) exhibit phenotypic plasticity. The foraging strategy of the individuals from the invader and resident species differed due to different evolutionary histories (Figure 1). Other than this difference, the only difference between individuals from the invader and resident species was their background death probability. Here we show results in which the invader has the default background death probability, equal to 0.001, and that of the resident increases from 0.001 (in which case there is no imposed competitive difference) to 0.01 (in which case the invader has a strong imposed competitive advantage). As expected, the invasion time decreased in both cases (PP and NPP) as the imposed competitive advantage of the invader was increased. Unexpected, however, is that invasion was much slower if both the resident and invader exhibited phenotypic plasticity than if they did not. Note that for a range of background death probability values, from 0.001 to about 0.0015 in which there was an imposed competitive advantage of the invader, there was no invasion with phenotypic plasticity, but there was invasion without phenotypic plasticity.
Figure 3.
Fitness surfaces for a focal consumer in the presence of either an optimal resident PP-consumer (A) or an optimal resident NPP-consumer (B). Each surface shows fitness as a function of the focal consumer foraging strategy, i.e., of the two traits that describe the probability of eating in predator presence and absence, respectively. Fitness is measured (Methods) in a habitat in which resource levels and dynamics have reached a steady state of the resident species. In (A) the resident PP-consumer has a probability of eating in predator presence and absence of 0.0 and 1.0, respectively (indicated by the plus sign [+]), which is the optimal value for a PP-consumer when alone. In (B) the resident NPP-consumer has a probability of eating of 0.45 in predator presence and absence (indicated by the asterisk [*]), which is the optimal strategy for an NPP-consumer when evolved alone. The fitness of a focal consumer, relative to the respective optimal individual in each figure, is given for each combination of eat probability traits (i.e., in predator presence and absence), where the isopleths in (A) and (B) represent fitness relative to the optimal strategies in the respective cases. For example, an individual with a 0.20 probability of eating in predator presence and 0.60 probability of eating in predator absence (indicated by the x ), had a fitness value of 0.996 with a resident PP-consumer (A) and 1.006 in a habitat with a resident NPP-consumer (B). Because NPP-consumers cannot respond differentially to predation risk, they are restricted to the diagonal (white) lines in (A) and (B), so that the probabilities of eating in predator presence and absence are always equal. Therefore NPP-consumers, when alone, will evolve to the highest point on that line as indicated in (B) by the asterisk (*). Figures (C) and (D) show fitness of the focal consumer as a function of probability to eat when that probability is the same in both predator presence and absence (i.e., along the diagonal in [A] and [B]), in habitats with resident PP-consumers (C) and NPP-consumers (D), respectively.
Figure 4.
Effect on Fitness of Deviation from Optimal Foraging Strategies
Curves show the average fitness of populations that deviate by a fixed Euclidean distance (averaged over a quarter circle for PP-consumers and two points for NPP consumers, see text) from the optimal strategy (i.e., optimal eat probability values), which are at the fitness peaks in Figure 2A (PP-consumers) and Figure 2B (NPP-consumers). Deviations in trait values from the optimum have a markedly larger negative effect on fitness for PP-consumers. Numbers indicate the proportional decrease in fitness for PP-consumers relative to NPP-consumers at different deviations in trait values from the optimal strategy. For example, PP-populations with trait values deviating from the optimum eat probability by 0.1 suffer a fitness reduction 7.8 times greater than NPP-populations with equivalent deviation from the optimum.
Figure 5.
Action Probability Parameters that Determine the Behavioral Strategy for a Single Individual PP-Consumer
There are two possible actions for each environmental state (i.e., different predation risks). In this example, the consumer has a 70% probability of performing the eat action (which can include moving to a neighboring cell) if the predator is absent, but 20% if the predator is present. Note that animals from the same species would have the same set of action probability parameters that determine the behavioral strategy, but the probability magnitudes may differ. Behavioral strategies with higher fitness will be selected and therefore “evolve” over time.