Mobility and its sensitivity to fitness differences determine consumer–resource distributions

An animal's movement rate (mobility) and its ability to perceive fitness gradients (fitness sensitivity) determine how well it can exploit resources. Previous models have examined mobility and fitness sensitivity separately and found that mobility, modelled as random movement, prevents animals from staying in high-quality patches, leading to a departure from an ideal free distribution (IFD). However, empirical work shows that animals with higher mobility can more effectively collect environmental information and better sense patch quality, especially when the environment is frequently changed by human activities. Here, we model, for the first time, this positive correlation between mobility and fitness sensitivity and measure its consequences for the populations of a consumer and its resource. In the absence of consumer demography, mobility alone had no effect on system equilibria, but a positive correlation between mobility and fitness sensitivity could produce an IFD. In the presence of consumer demography, lower levels of mobility prevented the system from approaching an IFD due to the mixing of consumers between patches. However, when positively correlated with fitness sensitivity, high mobility led to an IFD. Our study demonstrates that the expected covariation of animal movement attributes can drive broadly theorized consumer–resource patterns across space and time and could underlie the role of consumers in driving spatial heterogeneity in resource abundance.

The Jacobian matrix at positive equilibrium (obtained by setting all Eq.s S1-3 equal to 0) is: where ∆ = R H * − R L * and C T = (C H + C L )/2. From the above matrix, we can get the third-order polynomial of eigenvalue z: Using the Routh-Hurwitz criterion, and due to the fact that all the above coefficients of z 0 , z 1 , z 2 and z 3 > 0, any positive equilibrium (Eq. S1-4) is stable.
In what follows, we prove the uniqueness of the positive solution. By setting Eq. S1 and S2 = 0, we get: To ensure that the solution is positive, we must have R H * >0 and R L * >0, which is equivalent to We then let Eq. S3 = 0, that is, C L βe λ(w H −w L ) − C H βe λ(w L −w H ) = 0. After rearrangement, it becomes: where G represents a function of C H * . Because G decreases in C H * , so there exists at most one value C H * to make G(C H * ) = 0.
Replacing C L * by 2C T − C H * in S5 and rearranging the inequality, we get: 2CT - To ensure positivity of R H * and R L * , CT must satisfy: Based on (S7) and the uniqueness of a positive C H * in (S6), we must have G(2CT - > 0. Rearranging the above inequality, we get one necessary condition for G (2C T - Rearranging this inequality, we get necessary condition for G( In summary, necessary conditions of unique positive solution are (S8), (S9) and (S10), which indicated that, in general, to have positive R H * , R L * , C H * and C L * , total consumer abundance in the system (C T ) should not be too large, which would deplete resources (R L * = 0); C T also should not be too small, which would drive C L * to zero under fitness-dependent movement.

Appendix 2 The relationships among mobility, fitness sensitivity and the time for the system to reach equilibrium in the absence of consumer demography
In the absence of consumer demography, we used simulations to study how the time for the system to approach equilibrium depends on mobility and fitness sensitivity. Here, we define the solution approaching equilibrium when the density changes by less than 1e-6 within 10 continuous time-steps.
Without consumer movement between the two patches (i.e., β = 0), each patch would have its own equilibrium. The time to equilibrium depends on the initial densities of both consumers and resources in each patch. When consumers move (β > 0) but in random directions (i.e., no fitness sensitivity; λ = 0), the time to equilibrium depends on the density difference of consumers between the two patches. Here, we set initial densities of consumers to be equal in the two patches (i.e., no density difference of consumers), so there is no migration of consumers between the two patches when λ = 0. Therefore, in the above two scenarios (either β = 0 or λ = 0), the system has the same equilibrium and the same time to reach this equilibrium (the equilibrium time here is 53 steps; see the gray color line at β = 0 and λ = 0 in Fig. S1).
When consumers exhibit fitness-sensitive movement between the two patches (i.e., β > 0 and λ > 0), the time to equilibrium rapidly decreases as the baseline mobility increases (see the abrupt color change along β axis in Fig. S1). The time to equilibrium shows a unimodal pattern with respect to fitness sensitivity: i.e., when fitness sensitivity increases, the time to equilibrium first increases and then decreases. The unimodal pattern is stronger when the baseline mobility is relatively small (see the hump shape of time change along λ axis in Fig. S1). This unimodal pattern arises because when the fitness sensitivity becomes slightly larger than 0, the equilibrium changes: more consumers end up in the high-quality patch than in the low-quality patch (see Fig.   1a). For this simulation, the initial densities of consumers are equal in both patches, so the system needs more time to reach the new equilibrium. Once the fitness sensitivity increases up to a certain level, consumers can move to the high-quality patch faster, thus, the time to achieve equilibrium decreases. The smaller the baseline mobility is, the stronger the influence of fitness sensitivity on the system (i.e., the hump shape along λ axis is stronger when β is smaller in Fig.   S1).

Fig. S1
The relationships among mobility (β), fitness sensitivity (λ) and the time for the system to reach equilibrium when there is no consumer demography (p = 0). The color bar shows the time gradient to equilibrium: from gray to dark red, time increases. The parameters are: r H = 2, r L = 1, K H = 100, K L = 50, c = 0.05, α = 0.05, µ = 0.1 and C T =15. When ∆> 0 (which is true for our system), from (S14), we can get: ) and S15, we have dR H * dλ < 0 (S16) From S4 and S15, we have dC L * dλ < 0 (S17) From R L * = K L (− αC L * r L ) and S17, we have dR L * dλ > 0 (S18) Inequalities S15-S18 show that with the increase of λ, more consumers would move from low-quality patch to high-quality patch (C H * − C L * increases), and the disparity of resource densities would decrease (∆ = R H * − R L * decreases). This trend is always kept until ∆= 0 as λ → ∞. ∆= 0 is the limiting pattern under Ideal Free Distribution (IFD).

Appendix 4 The proof of the relationships between the fitness-sensitivity of consumers' movement and regional resource density in the absence of consumer demography
From Eq. S1-S2, we get R H