Rationalizing spatial exploration patterns of wild animals and humans through a temporal discounting framework

Understanding the exploration patterns of foragers in the wild provides fundamental insight into animal behavior. Recent experimental evidence has demonstrated that path lengths (distances between consecutive turns) taken by foragers are well fitted by a power law distribution. Numerous theoretical contributions have posited that “Lévy random walks”—which can produce power law path length distributions—are optimal for memoryless agents searching a sparse reward landscape. It is unclear, however, whether such a strategy is efficient for cognitively complex agents, from wild animals to humans. Here, we developed a model to explain the emergence of apparent power law path length distributions in animals that can learn about their environments. In our model, the agent’s goal during search is to build an internal model of the distribution of rewards in space that takes into account the cost of time to reach distant locations (i.e., temporally discounting rewards). For an agent with such a goal, we find that an optimal model of exploration in fact produces hyperbolic path lengths, which are well approximated by power laws. We then provide support for our model by showing that humans in a laboratory spatial exploration task search space systematically and modify their search patterns under a cost of time. In addition, we find that path length distributions in a large dataset obtained from free-ranging marine vertebrates are well described by our hyperbolic model. Thus, we provide a general theoretical framework for understanding spatial exploration patterns of cognitively complex foragers.

Rationalizing spatial exploration patterns of wild animals and humans through a temporal discounting framework

Understanding the exploration patterns of foragers in the wild provides fundamental insight into animal behavior. Recent experimental evidence has demonstrated that path lengths (distances between consecutive turns) taken by foragers are well fitted by a power law distribution. Numerous theoretical contributions have posited that “Lévy random walks”—which can produce power law path length distributions—are optimal for memoryless agents searching a sparse reward landscape. It is unclear, however, whether such a strategy is efficient for cognitively complex agents, from wild animals to humans. Here, we developed a model to explain the emergence of apparent power law path length distributions in animals that can learn about their environments. In our model, the agent’s goal during search is to build an internal model of the distribution of rewards in space that takes into account the cost of time to reach distant locations (i.e., temporally discounting rewards). For an agent with such a goal, we find that an optimal model of exploration in fact produces hyperbolic path lengths, which are well approximated by power laws. We then provide support for our model by showing that humans in a laboratory spatial exploration task search space systematically and modify their search patterns under a cost of time. In addition, we find that path length distributions in a large dataset obtained from free-ranging marine vertebrates are well described by our hyperbolic model. Thus, we provide a general theoretical framework for understanding spatial exploration patterns of cognitively complex foragers.

Broad-scale patterns of sex ratios in Patella spp.: a comparison of range edge and central range populations in the British Isles and Portugal

Sex change, or sequential hermaphroditism, occurs in the plant and animal kingdoms and often determines a predominance of the first sex. Our aim was to explore changes in sex ratios within the range of the species studied: Patella vulgata and Patella depressa. The broad-scale survey of sex with size of limpets covered a range of latitudes from Zambujeira do Mar (southern Portugal) to the English Channel. Indirect evidence was found for the occurrence of protandry in P. vulgata populations from the south of England, with females predominating in larger size-classes; cumulative frequency distributions of males and females were different; sex ratios were biased towards males and smallest sizes of males were smaller than the smallest sizes of females. In contrast in Portugal females were found in most size-classes of P. vulgata. In P. depressa populations from the south coast of England and Portugal females were interspersed across most size-classes; size distributions of males and females and size at first maturity of males and females did not differ. P. depressa did, however, show some indications of the possibility of slight protandry occurring in Portugal. The test of sex ratio variation with latitude indicated that P. vulgata sex ratios might be involved in determining the species range limit, particularly at the equatorward limit since the likelihood of being male decreased from the south coast of England to southern Portugal. Thus at the southern range limit, sperm could be in short supply due to scarcity of males contributing to an Allee effect.

Broad-scale patterns of sex ratios in Patella spp.: a comparison of range edge and central range populations in the British Isles and Portugal

Sex change, or sequential hermaphroditism, occurs in the plant and animal kingdoms and often determines a predominance of the first sex. Our aim was to explore changes in sex ratios within the range of the species studied: Patella vulgata and Patella depressa. The broad-scale survey of sex with size of limpets covered a range of latitudes from Zambujeira do Mar (southern Portugal) to the English Channel. Indirect evidence was found for the occurrence of protandry in P. vulgata populations from the south of England, with females predominating in larger size-classes; cumulative frequency distributions of males and females were different; sex ratios were biased towards males and smallest sizes of males were smaller than the smallest sizes of females. In contrast in Portugal females were found in most size-classes of P. vulgata. In P. depressa populations from the south coast of England and Portugal females were interspersed across most size-classes; size distributions of males and females and size at first maturity of males and females did not differ. P. depressa did, however, show some indications of the possibility of slight protandry occurring in Portugal. The test of sex ratio variation with latitude indicated that P. vulgata sex ratios might be involved in determining the species range limit, particularly at the equatorward limit since the likelihood of being male decreased from the south coast of England to southern Portugal. Thus at the southern range limit, sperm could be in short supply due to scarcity of males contributing to an Allee effect.