Uncertainty in foraging success and its consequences on fitness
Introduction
Optimal foraging models are an important tool in studying the foraging strategies of organisms. These models describe the relationships between foraging strategies and their consequences on fitness; the strategy that results in the highest fitness is considered as optimal (Stephens and Krebs, 1986). Theoretical and empirical studies of foraging behavior commonly use foraging success (e.g., rate of energy intake) as a proxy for fitness (e.g., Charnov, 1976a, Charnov, 1976b; Mori and Boyd, 2004; Semmler et al., 2021) described by the following relationship:
foraging strategy → foraging success → fitness,
where the direction of arrow indicates the cause-and-effect relationship (e.g., foraging strategy has a causal effect on foraging success). In other words, a model may describe the relationship between foraging strategy and its consequences for foraging success. Meanwhile, foraging success is assumed to be highly correlated with fitness such that the foraging strategy that maximizes foraging success also maximizes fitness.
Although foraging success is commonly used as a proxy for fitness, the strategies maximizing foraging success and fitness are not necessarily the same when (1) the relationship between foraging success and fitness is nonlinear and (2) foraging success is associated with uncertainty (Okuyama, 2020). Empirical studies revealed linear and nonlinear relationships between foraging success (e.g., amount of food consumption or body size) and fitness (reproductive success) (Cobo and Okamori, 2008, Gao et al., 2016, Linhares et al., 2014, Wootton, 1977). However, these relationships are not applicable for foragers at non-reproductive states (e.g., juveniles). When the relationships between daily foraging success and expected fitness for non-reproductive stages are estimated by models, the key factor is a variable that is influenced by daily foraging success and affects daily survival (Okuyama, 2020). For example, the fat reserve of birds may increase or decrease daily depending on their foraging success (Lilliendahl, 2002, Pravosudov and Grubb, 1998) and influences their daily survival (Houston and Mcnamara, 1993). In this case, the relevant time span for evaluating foraging success is one day (although one day is used as an example, it can be any relatively short time spans). In the presence of such variable, nonlinearity between foraging success and expected fitness for non-reproductive stages is ensured regardless of the relationship at the time of reproduction (Okuyama, 2020). Meanwhile, if daily foraging success has no influence on daily survival, then the relationship between foraging success and expected fitness for non-reproductive stages is primarily determined by their association at the time of reproduction, and foraging success may be evaluated for the entire pre-reproductive duration.
The second condition (i.e., foraging success is associated with uncertainty) is the norm and is nearly always satisfied to some degree. Even when the strategy is completely known, the foraging success associated with a given strategy cannot be predicted with certainty because of the influence of luck (e.g., probabilistic prey encounter) (Wilson et al., 2018). For example, the foraging success (the number of prey captured) of an orb web spider cannot be predicted with certainty even when the characteristics of the web and the density of prey in the field are known (da Silva et al., 2021). In behavioral ecology, risk-sensitive foraging theory explicitly considers uncertainty in foraging success (McNamara and Houston, 1992, Stephens and Krebs, 1986). The theory shows that the aforementioned two conditions (i.e., nonlinear fitness function and uncertainty in foraging success) may induce Jensen’s inequality (Stephens and Krebs, 1986), which states that when w is a convex function and when w is a concave function. In this formula, X is a random variable, and and are the means of X and respectively. In the case of foraging, X represents the stochastic foraging success (e.g., the number of prey captured), and the function w represents the relationship between foraging success and fitness. However, the influence of risk-sensitive foraging on general foraging problems is largely unknown. Therefore, this study examines the effect of uncertainty in foraging success on fitness by considering Jensen’s inequality.
The two conditions depend on the fact that foraging success is evaluated in a finite duration. If the foraging duration is indefinitely long, then no uncertainty will be detected for the foraging success (a specific example will be given below). On this basis, this study investigated the effect of finite foraging duration. Optimal foraging models that explicitly considered finite foraging durations have been developed (e.g., Luttbeg et al., 2020; Wajnberg et al., 2006; Watts et al., 2018), but the details treated in this work are usually not incorporated in these models. Although it may be trivial that foraging strategy is stochastic and uncertain, the actual characteristics of uncertainty in foraging success are poorly understood. The effect of uncertainty in foraging success cannot be studied unless the probability distribution of all possible foraging outcomes can be described for all feasible strategies.
This study illustrates how uncertainty in foraging success associated with a foraging strategy affects the fitness value of the strategy. A well-known prey choice model (Charnov, 1976a) was selected as an example (referred to as the original model). The original model derived the relationship between foraging strategies and their consequences on long-term energy intake rate, that is, the rate of energy intake (i.e., foraging success) is the proxy for fitness. A stochastic simulation model was used to characterize the probability distributions of foraging outcomes associated with foraging strategies. As mentioned above, foraging duration is assumed to be limited in the simulation model. When foraging duration in the simulation model becomes indefinitely long, the simulation and original models generate the same predictions.
Section snippets
Model
The model considers the predation scenario addressed in the well-known prey choice model (Charnov, 1976a). A predator’s strategy is described by a set of probabilities (p1, p2, …, pn), where pi is the probability of attacking prey type i upon encounter. For simplicity, this study considers only two prey types such that a strategy is defined as a set (p1, p2). Predators eating type i prey spend a duration of hi (handling time) and gain some energy ei (i = 1, 2). The profitability of a prey is
Results
The effect of foraging duration on foraging success was illustrated in a simple scenario where predators only consume primary prey (λ2 = 0). In the example (Fig. 1a), the encounter rate with primary prey λ1 is 1 (i.e., the expected number of primary prey encountered in 1 h is 1), and the handling time h1 is 1 h. The expected rate of energy intake predicted by Eq. (1) is 0.5 when e1 = 1. The expected energy intake rate estimated by the simulation model generally matches with Eq. (1) when the
Discussion
Foraging strategy influences the expected foraging success and the uncertainty in foraging success. In this study, stochastic encounter with prey is the source of uncertainty, and this uncertainty can be regarded as luck. The effect of luck diminishes over time because an individual is unlikely to be consistently lucky or unlucky for a long duration. In the presence of uncertainty in foraging success, the strategies maximizing the expected foraging success and the fitness may not be the same
Declaration of Competing Interest
None declared.
Acknowledgments
I thank two anonymous reviewers for their constructive comments. This study was supported by the Ministry of Science and Technology (Taiwan) (grant number 108–2311-B-002–017-MY3).
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