An increase in search time S1 will likely make prey2:
As S1 increases, the predator must wait longer to encounter the preferred prey 1. The expected rate from rejecting prey 2 and continuing to search therefore declines, making the immediate lower-profitability item more acceptable. This is the central diet-breadth prediction of optimal foraging theory: scarcity of the highest-ranked prey broadens the optimal diet, whereas abundance permits specialization. The profitability E2/h2 of prey 2 need not change; what changes is its value relative to the opportunity cost of continued searching. An increase in search time consequently shifts the acceptance threshold toward inclusion rather than making prey 2 less relevant. This reasoning connects organismal behavior with broader ecological consequences. A mechanism that changes encounter rates, resource use, or reproductive success can scale up to alter population trajectories and the structure of species interactions. The alternatives can be separated by asking what changes for each participant and which process causes that change. That approach is more reliable than treating familiar examples as fixed labels, because many interactions shift with environmental context.
Ref: Ecology: From Individuals to Ecosystems, Begon et al., 5th Ed., Ch. 10