Fig. 2.

Scaling the gains and invariance. Function (21) is used as an illustration of class (23). a For homogeneous habitats, holding travel time fixed at $T$, $t^{\ast }$ is invariant to quality (green vertical line). b $t_i^{\ast }$ always increases with $x_i$ (vertical arrows), except in the homogeneous case (dot on the far right). In this example $x_2$ and $p_2$ were varied in a habitat with two other patch-types, having qualities $x_1=1$ and $x_3=3$, and relative frequencies $p_1=p_3=(1-p_2)/2$. Remark that increasing $p_2$ increases $t_2^{\ast }$ if it decreases $E_n^{\ast }$ (low $x_2$ values, below dotted curve) and decreases $t_2^{\ast }$ otherwise (high $x_2$ values, above dotted curve). c Except in the homogeneous case (black line), the average residence time varies with $x_2$. It increases for low $x_2$ values and decreases for high $x_2$ values. d In a heterogeneous habitat, if all gain functions are scaled by the same factor, optimal residence times do not vary. In this example, all $x$ values where multiplied by $\frac{5}{4}$ (from gray to black). Other parameters: $T=1$, $\mathit{\lambda }=1$ (color figure online)