Mathematical Models of Fertilization—An Eco-Evolutionary Perspective

Abstract

Mathematical models of fertilization have been developed for many taxa, for both external and internal fertilizers. They estimate the proportion or number of fertilized gametes based on gamete concentrations and parameters relating to the biology of the model organism, as well as serve multiple purposes: a predictive purpose, with applications in, for example, artificial insemination; they clarify causal components of fertilization success such as concentration, size, collision rates and swimming speed of gametes, and polyspermy block times; and they function as components of models in evolutionary ecology, which often require understanding of fitness consequences of resource allocation between gametes and other traits. We pay particular attention to this last category, which has received less attention than other uses. Many evolutionary models assume the simplest relationship between fertilization success and gamete numbers: all eggs are fertilized. In nature, however, it is not uncommon for eggs to be sperm limited, and fertilization success must decrease as sperm density approaches zero. Fertilization functions become important in the range between these two extremes. We focus on models in evolutionary ecology, but aim for a resource that is useful regardless of topic or taxon by reviewing models developed for different purposes in a common mathematical framework.

Evolution of Anisogamy in Organisms with Parthenogenetic Gametes

“…This symmetrical fertilization function has been derived in previous work for the purpose of modeling fertilization for the entire range from isogamy to extreme anisogamy (Togashi et al 2007;Lehtonen 2015;Lehtonen and Dardare 2019), which makes it particularly suitable for a model of anisogamy evolution. The parameter a can be used to adjust fertilization efficiency to simulate scenarios from extreme gamete limitation to ones where all gametes of the less numerous type are fertilized.…”

Sexual selection after gamete release in broadcast spawning invertebrates

“…As we highlight here, theoretical models incorporating the distinctive dynamics of externally fertilizing systems have made considerable progress in bridging this divide (e.g. see review in [20]).…”

The Legacy of Parker, Baker and Smith 1972: Gamete Competition, the Evolution of Anisogamy, and Model Robustness

“…Under such group-structured conditions, gamete competition and gamete limitation can both select for increased gamete numbers simultaneously, and they tend to act in the same direction, reinforcing each other (see [ 8 ] and Figure 1 therein). Several mathematical models accounting for fertilisation success under various assumptions have been developed (reviewed in [ 37 ]) which can be used to model the components and and thus include selection via both gamete competition and gamete limitation in the same model.…”