Rapid clade divergence and phyletic gradualism in an interacting particle model of sympatric speciation.

The coexistence of cladogenesis, i.e., the branching of lineages along an evolutionary tree as observed in the fossil record, and anagenesis, which is the progressive evolution within populations, lacks a clear explanation. In this study, we examine a simple model that simulates the evolutionary changes occurring within populations inhabiting the same environment in sympatry, and driven by ecological competition. Our model characterizes populations through a set of evolving morphological traits represented by mathematical points within a two-dimensional morphospace. Such points may reproduce or die due to overcrowding, implying competition in morphospace as suggested by the ecological phenomenon of character displacement. By focusing on the morphospace rather than physical space, the model effectively captures the simultaneous evolution of coexisting populations. Central to the model is the delicate balance between the range of competition and the range of reproduction within the morphospace. Interesting patterns emerge when the ratio between the competition to reproducetion ranges, referred to as CR ratio, changes from values slightly smaller to significantly larger than unity. When competition acts over short distances relative to the reproduction range (low CR), the phylogenetic tree takes on a nearly uniform appearance, gradually transforming into a more bush-like structure for slightly higher CR values. With further increases in CR, evolutionary lineages become more discernible, and the morphogenetic pattern shifts from a bush-like shape to a more tree-like arrangement and few branches for very large CRs. At specific time sections, the synthetic phylogenetic tree appears as an assembly of clusters of individuals within the morphospace. These clusters, interpretable as simulated models of species, exhibit distinct separation within the morphospace and are subject to dynamic inter-cluster repulsion. Notably, clusters tend to be resistant to change. They maintain relatively constant abundances while gradually shifting their positions within the morphospace-a phase that aligns with the concept of phyletic gradualism. However, this predictable pattern is occasionally upset by the abrupt divisions into multiple groups, interpreted as cladogenesis events. The intricacies of the splitting process are explored, revealing that in scenarios with large CR values, the splitting can emerge much more rapidly than phyletic changes. This accelerated process of splitting is initiated by one or few individuals at the fringes of a cluster, where competition is minimal. The newly generated cluster then undergoes deformation, swiftly followed by divergence and splitting (seen as branching in the synthetic phylogenetic tree), as if an inherent "repulsion" triggered the division between species. The simple rules implied in the interacting-particle model may provide insight into the coexistence of gradualism and cladogenesis along lineages, illustrating the capacity for rapid shifts during cladogenesis and the more gradual process of anagenesis.