On invasion boundaries and the unprotected coexistence of two strategies

Priklopil, Tadeas

doi:10.1007/s00285-011-0448-y

Cited by 14 publications

(16 citation statements)

References 25 publications

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“…Two theorems byPriklopil (2012) connect the unfolding of M = 0 with the shortterm population dynamics of two coexisting resident strategies as follows (seePriklopil (2012) for details). (i) For > 0 sufficiently small, the invasion boundaries of the pairwise invasibility plot generically intersect in the neighbourhood of the singularity either when M = or when M = − .…”

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confidence: 99%

Construction of multiple trade-offs to obtain arbitrary singularities of adaptive dynamics

Kisdi

2014

J. Math. Biol.

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Evolutionary singularities are central to the adaptive dynamics of evolving traits. The evolutionary singularities are strongly affected by the shape of any trade-off functions a model assumes, yet the trade-off functions are often chosen in an ad hoc manner, which may unjustifiably constrain the evolutionary dynamics exhibited by the model. To avoid this problem, critical function analysis has been used to find a trade-off function that yields a certain evolutionary singularity such as an evolutionary branching point. Here I extend this method to multiple trade-offs parameterized with a scalar strategy. I show that the trade-off functions can be chosen such that an arbitrary point in the viability domain of the trait space is a singularity of an arbitrary type, provided (next to certain non-degeneracy conditions) that the model has at least two environmental feedback variables and at least as many trade-offs as feedback variables. The proof is constructive, i.e., it provides an algorithm to find trade-off functions that yield the desired singularity. I illustrate the construction of trade-offs with an example where the virulence of a pathogen evolves in a small ecosystem of a host, its pathogen, a predator that attacks the host and an alternative prey of the predator.

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confidence: 99%

Construction of multiple trade-offs to obtain arbitrary singularities of adaptive dynamics

Kisdi

2014

J. Math. Biol.

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“…The resident-mutant coex-6 istence region is locally a cusp rooted at the singular point (x,x) (see [Priklopil, 2012, Dercole & Geritz, 2015), and though there might generically be up to two coexistence equilibria, only one is stable and 8 should be considered for developing a proper expansion of the dimorphic fitness. Further research could investigate the codimension-two bifurcation at which both fitness second-derivatives vanish (the type of 10 coexistence is already available in [Dercole & Geritz, 2015]), or the cases at whichλ (0,3) 1 = 0 together with one of the fitness second-derivatives; or, as well, higher codimensions that do occur in applications…”

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confidence: 99%

The Branching Bifurcation of Adaptive Dynamics

Rossa

Dercole

Landi

2015

Int. J. Bifurcation Chaos

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We unfold the bifurcation involving the loss of evolutionary stability of an equilibrium of the canonical equation of Adaptive Dynamics (AD). The equation deterministically describes the expected long-term evolution of inheritable traits-phenotypes or strategies-of coevolving populations, in the limit of rare and small mutations. In the vicinity of a stable equilibrium of the AD canonical equation, a mutant type can invade and coexist with the present-resident-types, whereas the fittest always win far from equilibrium. After coexistence, residents and mutants effectively diversify, according to the enlarged canonical equation, only if natural selection favors outer rather than intermediate traits-the equilibrium being evolutionarily unstable, rather than stable. Though the conditions for evolutionary branching-the joint effect of resident-mutant coexistence and evolutionary instability-have been known for long, the unfolding of the bifurcation remained a missing tile of AD, the reason being related to the nonsmoothness of the mutant invasion fitness after branching. In this paper, we develop a methodology that allows the approximation of the invasion fitness after branching in terms of the expansion of the (smooth) fitness before branching. We then derive a canonical model for the branching bifurcation and perform its unfolding around the loss of evolutionary stability. We cast our analysis in the simplest (but classical) setting of asexual, unstructured populations living in an isolated, homogeneous, and constant abiotic environment; individual traits are one-dimensional; intra-as well as inter-specific ecological interactions are described in the vicinity of a stationary regime.

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“…As we unfold a model with M = 0 into a model with nonzero M, we encounter a situation where the invasion boundaries generically intersect each other (cf. Figure 4); and in the neighbourhood of such an intersection point, coexistence occurs also outside the area of mutual invasibility (Priklopil 2012). Hence the construction method yields not only examples for evolutionary branching, but also examples of unprotected coexistence with multiple attractors in the population dynamics.…”

Section: Discussionmentioning

confidence: 94%

“…where in the last step we substituted dN (α * ) ds from equation (46), using (41) with s 0 = g 0 (P * ) P * 1 − P * S * Ŝ (α * ) P (α * ) (cf. (35). In (49), the derivativesN (α * ),Ŝ (α * ) andP (α * ) are from equations (28), (29) and (31), respectively.…”

Section: Transversality Condition For Unfolding M =mentioning

confidence: 99%

A construction method to study the role of incidence in the adaptive dynamics of pathogens with direct and environmental transmission

Kisdi

Boldin

2012

J. Math. Biol.

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We study the adaptive dynamics of virulence of a pathogen transmitted both via direct contacts between hosts and via free pathogens that survive in the environment. The model is very flexible with a number of trade-off functions linking virulence to other pathogen-related parameters and with two incidence functions that describe the contact rates between hosts and between a host and free pathogens. Instead of making a priori particular assumptions about the shapes of these functions, we introduce a construction method to create specific pairs of incidence functions such that the model becomes an optimization model. Unfolding the optimization model leads to coexistence of pathogen strains and evolutionary branching of virulence. The construction method is applicable to a wide range of eco-evolutionary models.

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On invasion boundaries and the unprotected coexistence of two strategies

Cited by 14 publications

References 25 publications

Construction of multiple trade-offs to obtain arbitrary singularities of adaptive dynamics

Construction of multiple trade-offs to obtain arbitrary singularities of adaptive dynamics

The Branching Bifurcation of Adaptive Dynamics

A construction method to study the role of incidence in the adaptive dynamics of pathogens with direct and environmental transmission

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