Spatial memory shapes density dependence in population dynamics

Most population dynamics studies assume that individuals use space uniformly, and thus mix well spatially. In numerous species, however, individuals do not move randomly, but use spatial memory to visit renewable resource patches repeatedly. To understand the extent to which memory-based foraging movement may affect density-dependent population dynamics through its impact on competition, we developed a spatially explicit, individual-based movement model where reproduction and death are functions of foraging efficiency. We compared the dynamics of populations of with- and without-memory individuals. We showed that memory-based movement leads to a higher population size at equilibrium, to a higher depletion of the environment, to a marked discrepancy between the global (i.e. measured at the population level) and local (i.e. measured at the individual level) intensities of competition, and to a nonlinear density dependence. These results call for a deeper investigation of the impact of individual movement strategies and cognitive abilities on population dynamics.


Effect of the spatial clustering of resource patches
We simulated an environment with patches aggregated in super-patches following Benhamou (1992) 1 .
The same number of identical resource patches as in the main analyses (Np = 400) were distributed among 20 clusters, with a within-cluster patch density of 0.15 (Fig. S1). Running the model on this spatially clustered environment did not qualitatively change our results: the carrying capacity of populations of memory users was still much higher than that of non-memory users, and both carrying capacities were close to that in the non-clustered case (Fig. S2), and memory use led to heavy environmental depletion (Fig S3), to smaller local intensities of competition for high global population densities (Fig S4), and to nonlinear density-dependence (Fig S5-6).
1 Benhamou (1992). Efficiency of area-concentrated searching behaviour in a continuous patchy environment. Journal of Theoretical Biology. 159:67-81.    For populations of with-memory individuals, the best-fitting model is the piecewise 2 nd -order polynomial regression (all ΔAIC > 500), with an estimated breakpoint occurring at a population size of 13. At this population size, the negative slope of density-dependence is multiplied by more than 10. Sensitivity of the carrying capacity and of the shape of the density-dependence of populations of with-memory individuals to the durations of working (T W ) and reference (T R ) memories.

Carrying capacity
The carrying capacity of populations of with-memory individuals is very robust to changes in the memory parameters TW and TR. For TR ranging between 300 and 1200 and For TW ranging between 200 and 1000 (with TW <TR), the carrying capacity varies between 15.1 and 18.2 (Fig. S7). The black star represents the default parameter configuration.

Shape of the density-dependent response curve
We consider that a breakpoint is present in the density-dependence response curve if the difference in AIC between the with-breakpoint model fitted by nonlinear least squares and the no-breakpoint models is greater than 2. In such cases, we estimate the population size at which the breakpoint occurs and the increase in negative slope that occurs at the breakpoint. When varying the memory parameters TW between 200 and 1000 and TR between 300 and 1200 (with TW <TR), all estimation procedures converge, except for a few combinations of parameters (TW ; TR): (300 ; 700), (600 ; 800), (700 ; 1200), (800 ; 1000), and (800 ; 1200), for which no breakpoint is visually present (see Fig. S9). When the model converges, the fit of the segmented polynomial model is better than a simple polynomial, a Beverton-Holt, a Ricker or a theta-logistic model (all ΔAIC > 2) for all combinations of memory parameters. The location of the breakpoint, when it exists, is very robust and occurs on average for population sizes of 14.5 + 0.8. At the breakpoint, the slope always increases in absolute value; the ratio of slopes after vs before the breakpoint ranging between 1.2 and 14.7, with a mean of 4.8 + 3.5 ( Fig S8). The increase of the slope at the breakpoint depends mostly on TW, with smaller values of TW leading to a sharper increase in the strength of the negative density-dependence at the breakpoint.  Note that a small Vthr value (left panel) leads to smaller per capita growth rates because with-memory individuals go back to poor or distant known patches more frequently than when the threshold is larger, thus decreasing their foraging efficiency. On the other hand, as there is a range of Vthr values for which individuals can always, after a learning phase, improve their intake rate (cf Riotte-Lambert et al. 2015 2 ), increasing Vthr within this range always lengthens the learning phase, but also improves the performance of individuals on the long run because they use only the best patches. This explains why for a larger Vthr the population dynamics of with-memory individuals diverges even more from that of populations of memoryless individuals (right panel).

Supporting Information 4
Sensitivity of the carrying capacity and of the shape of the density-dependence to the energetic cost of movement.
With all other parameters set to default values, for an energetic cost of movement equal to twice the default value, 100% of populations of without-memory individuals and 76% of populations of with-memory individuals go extinct. The remaining populations of this latter type do not grow and stay at a mean population size of 1.96 (+ 1). For an energetic cost of movement equal to half the default value, the mean + SD carrying capacity increases to 25.78 + 0.4 (with memory) and 25.07 + 0.54 (without memory). At the end of the simulation, the mean + SD intake rate is similar for with-and without-memory individuals (2.32 + 0.17 vs 2.23 + 0.10).   S12 shows two examples of the spatial patterns for the last time window of the simulation of a population of with-memory individuals with a cost of movement set to default value vs half the default value (left and right panels). It appears that individuals are more numerous and each uses larger areas when the cost of movement is low, which leads to an increased overlap. Sensitivity of the carrying capacity and the shape of the density-dependence to the value of the energetic threshold that an individual must reach to reproduce.
The carrying capacity of the two types of populations is robust to changes in the difference between the energetical state Erep that an individual must reach to reproduce and the value to which its state is lowered following a reproduction event (Fig. S13). The carrying capacity is equal to 16.67 + 0.19 (with memory) and 11.45 + 0.41 (without memory) for a halved difference (Erep=375), and is equal to 16.11 + 0.35 (with memory) and 11.20 + 0.50 (without memory) for a doubled difference (Erep=750).

Fig. S13.
Mean + SD (between simulations) per capita growth rate as a function of total population size for both types of populations (with-memory: plain line and grey area; without memory: dashed line and zebra area). All parameters were set to default values, except Erep, which was set to 375 (left panel) and 750 (right panel). The red line represents the predicted values of the piecewise 2 nd -order polynomial regression fitted to populations of memory users. The estimated location of the breakpoint is indicated by an orange arrow.
For Erep=375, the shape of density-dependence for populations of with-memory individuals shows a breakpoint (all ΔAIC > 100 with the no-breakpoint 2 nd -order polynomial, the Beverton-Holt, Ricker and theta-logistic models) estimated at a population size of 14. On the left side of this point, the slope of density-dependence is slightly positive (0.008), and on the right side, the slope abruptly becomes negative (-0.05). For Erep=750, a breakpoint is also present (all ΔAIC > 80) and occurs at an estimated population size of 15. At this population size, the slope of density-dependence is 3.7 times greater.
Note that for Erep=750, we ran the simulations for 400,000 time steps instead of 200,000 to allow for the population to reach its carrying capacity.