Cooperation under predation risk: a data-based ESS analysis

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Abstract
Two fish that jointly approach a predator in order to inspect it share the deadly risk of capture depending on the distance between them. Models are developed that seek ESS inspection distances of both single prey and pairs, based on experimental data of the risk that prey (sticklebacks) incur when they approach a predator (pike) to varying distances. Our analysis suggests that an optimal inspection distance can exist for a single fish, and for two equal fish behaving entirely cooperatively so as to maximize the fitness of the pair. Two equal fish inspecting cooperatively should inspect at an equal distance from the predator. The optimal distance is much closer to the predator for cooperative pairs than for single inspectors. However, optimal inspection for two equal fish behaving cooperatively operates across a rather narrow band of conditions relating to the benefits of cooperation. Evolutionarily stable inspection can also exist for two equal fish behaving non–cooperatively such that each acts to make a best reply (in terms of its personal fitness) to its opponent's strategy. Non–cooperative pairs should also inspect at equal distance from the pike. Unlike the ‘single fish’ and ‘cooperative’ optima, which are unique inspection distances, there exists a range of ESS inspection distances. If either fish chooses to move to any point in this zone, the best reply of its opponent is to match it (move exactly alongside). Unilateral forward movement in the ‘match zone’ may not be possible without some cooperation, but if the pair can ‘agree’ to move forward synchronously, maintaining equal distance, inspection will occur at the nearest point in this zone to the predator. This ‘near threshold’ is an ESS and is closer to the predator than the single fish optimum—pairs behaving almost selfishly can thus attain greater benefits from inspection by the protection gained from Hamilton's dilution effect. That pairs should inspect more closely than single fish conforms with empirical findings. Phenotypic asymmetries in costs and benefits between the fish are not yet included in the model.