Dynamic optimization over infinite-time horizon: Web-building strategy in an orb-weaving spider as a case study

Dynamic state-dependent models have been widely developed since 1990s for solving questions in evolutionary ecology. Up to now, these models were mainly run over finite-time horizon. However, for many biological questions an infinite-time horizon perspective could be more appropriate, especially when the end of the modeled period is state- rather than time-dependent. Despite this approach is widely used in the field of economics and operational research, thus far no work has been providing biologists with a general method to solve infinite-time horizon problems. Here we present such a method, through the exhaustive description of an algorithm that we implement to determine the strategy an organism should follow to reach a particular state as fast as possible while limiting mortality risk. To illustrate that method we explored web-building behavior in an orb-weaving spider. How are adult females predicted to build their successive webs to gain energy, grow, and lay their first clutch as fast as possible, without suffering from either predation or starvation? From this example, we first show how an optimal strategy over infinite-time horizon can be processed and selected. Second, we analyse variations of the optimal web-building strategy along with the spider's body weight and predation risk during web building. Our model yields two main predictions: (1) spiders reduce their web size as they are gaining weight due to body-mass-dependent cost of web-building behavior, and (2) this reduction in web size starts at lower weight under higher predation risk.