Coevolution of cannibalistic predators and timid prey: evolutionary cycling and branching

Interactions Obtained from Basic Mechanistic Principles: Prey Herds and Predators

“…When m > s, the isocline in (27) intersects the vertical axes at y = 1 − m s < 0 and we find at most one interior equilibrium E * . We obtain the feasibility condition for E * by imposing that the curve in (27) takes positive values at x = 1, that is, if m < s + ea.…”

“…Note that the isocline in (26) intersects the x-axis at (0, 0) and (1, 0) and has a maximum at x = 1−α 2−α < 1 2 for 0 < α < 1, while the isocline in (27) intersects the vertical axis at (0, 1 − m s ) and is a root function translated by 1 − m s and dilated by ea s . Therefore, if the intersection point of the isocline in (27) lies in the positive quadrant, i.e., if m < s, we find three different configurations for the phase plane: the two isoclines can intersect at most twice at…”

Extinction in complex communities as driven by adaptive dynamics

“…Amending population dynamics equations to include such adaptive changes to interspecies interactions is referred as ‘adaptive dynamics’ (AD) (Abrams, 1999; Abrams, 2000; Cortez et al, 2020; Dieckmann & Law, 1996; Dieckmann et al, 1995; Gandon et al, 2008; Gavrilets, 1997; Hochberg & Holt, 1995; Jian et al, 2016; Lehtinen & Geritz, 2019; Lion, 2018; Loeuille & Hauzy, 2018; Marrow et al, 1996; Rosenzweig et al, 1987; Rosenzweig & Schaffer, 1978; Schaffer & Rosenzweig, 1978). We should caution that AD is applicable only to asexual, well‐mixed communities, in the low mutation rate limit.…”

“…As with much of classical population dynamics, AD typically focuses on demonstrating the stability of communities—even in the face of perpetual evolutionary arms races (Cortez & Patel, 2017; Cortez et al, 2020; Jian et al, 2016; Lehtinen & Geritz, 2019; Weitz et al, 2005). However, adaptive changes in interspecies interactions can occasionally lead to catastrophic displacements in equilibrium abundances, and even extinction, as was suggested theoretically (Boldin & Kisdi, 2016; Gyllenberg & Parvinen, 2001; Leimar, 2002; Marrow et al, 1996; Matsuda & Abrams, 1994; Matsuda & Abrams, 1994; Parvinen, 2005; Parvinen, 2010; Parvinen & Dieckmann, 2013) and empirically (Conover & Munch, 2002; Fiegna & Velicer, 2003; Howard et al, 2004; Muir & Howard, 1999; Olsen et al, 2004; Rankin & López‐Sepulcre, 2005).…”

“…While standard LV equations keep A ij unchanging, AD considers parameters that depend on some traits . The specific functional form of depends on the ecological details of the system of interest (Abrams, 1999; Abrams, 2000; Cortez et al, 2020; Dieckmann & Law, 1996; Dieckmann et al, 1995; Gandon et al, 2008; Gavrilets, 1997; Hochberg & Holt, 1995; Jian et al, 2016; Lehtinen & Geritz, 2019; Lion, 2018; Loeuille & Hauzy, 2018; Marrow et al, 1996; Rosenzweig et al, 1987; Rosenzweig & Schaffer, 1978; Schaffer & Rosenzweig, 1978). However, the defining aspect of all AD models is that novel strains with differing traits (and therefore different interaction matrix elements) will be introduced, and subsequently, LV equations will govern the fate of these strains, thereby playing the role of natural selection.…”