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## Spatially explicit models with adaptation

As a mode of example Kirkpatrick and Barton’s model (Kirkpatrick and Barton 1997) studies the dynamics of population density and mean trait value of a population structured by a phenotypic trait. This model supposes that there is an optimal trait value θ(x) = Bx such that when the phenotypic trait z of an individual differs from this optimal, there is a penalization in fitness. The dynamics of the model are given Here, the parameter A represents the adaptation potential of the considered population, and B is a measure of the degree of change of the phenotypical cline, so it is ameasure of spatial heterogeneity. Before moving on to the principal results of Kirkpatrick and Barton (1997), observe that equation (1.6b) corresponds to a spatial diffusion term, corrected by population density, plus a local selection term, which drives population towards the optimum trait value. This term is standard in quantitative genetics, and it is directly proportional to the derivative of the absolute fitness with respect to the phenotypic trait (Lande 1976, 1979): δz = h2 ∂r ∂z = h2 ∂r ∂z 1 − n − 1 2 (¯z − Bx)2 = −2h2 (¯z − Bx) .

### Evolutionary invasion speeds and invasion mechanisms

Current global changes make it urgent to understand the eco-evolutionary determinants on species’ ranges. A deep understanding of the way populations can help us derive efficient and useful conservation and control policies. This is also useful for understanding the impacts on biodiversity, as species modify their ranges and thus their interaction with other species.

Usual models used to describe distributions are only based on measurements of climatic variables, which do not provide a mechanism explaining them. These models do not consider adaptation to environment either, which has been observed to be an important determinant on geographic distributions, even at short timescales. We take Kirkpatrick and Barton’s model as a starting point, which is a monospecific model accounting for adaptation to environment and spatial heterogeneity. It assumes that individuals are characterized by a phenotypical trait and that there is a cline for optimal values of this phenotype in space, so that deviation from the optimal implies a fitness penalty. If we denote the space by x, then the optimal cline is given by θ(x) = Bx. Denoting the density of individuals by n(t, x) and the mean trait of the considered population by z(t, x), the dynaics are then given by the equation system.

#### Effects of predation on evolutionary invasion speeds and species distributions

We would also like to understand the way interspecific interactions modify the way species and populations establish themselves along space. Usual models used to describe distributions do not always consider interspecific interactions, and even if they do, they do not consider possible fast adaptations. The model by Kirkpatrick and Barton (1997) is a good starting point for understanding one-species distributions along a linear gradient, but it ignores interspecific interactions. Although there have been studies considering competing species along an environmental gradient Case and Taper (2000) and Norberg et al. (2012), the predation case, for simultaneously adapting prey and predator, has not yet been studied. There is evidence of interactions between community and evolutionary context in empirical studies, highlighting the importance of simultaneously considering interactions and adaptation.

Here, we propose a spatially explicit model accounting for predator-prey interactions and adaptation to environment along an optimal phenotypic cline. We consider two interacting prey and predator populations distributed along onedimensional space, parameterized by the variable −1 < x < 1, with dynamics taking place over time t 0. We suppose that populations are structured by a phenotypic variable −1 < z < 1 and that, for each population, there is an optimal phenotypic trait value θ1(x) = b1x (for prey) and θ2(x) = b2x (for predators).

**Relating propagation speeds to adaptation regimes**

A natural question is then how these extreme scenarios relate to Kirkpatrick and Barton’s one-species model in terms of propagation speeds, which provides a method to concretely determine what strong and weak adaptations mean. To understand this, we numerically approximate the solution of the one-species model for a variety of parameter pairs (A and B) and compare the results to those given by the extreme adaptation limits (Figure 2.3. Refer to the Appendix 2.6.2 for details on the numerical scheme).

**Effects of dispersal and adaptation potential on the evolution of species ranges in a predator-prey framework**

While single-species models in the line of Kirkpatrick and Barton 1997 andM´endez- Vera et al. 2019 highlight the effect of varying adaptation potential for a given species, they do not account for the fact that the community context may also vary along the environmental cline. The density of enemies may for instance vary, affecting species density, thereby modulating local adaptation and affecting the distribution of gene flows. To tackle this part of the question, we turn to a predator-prey model in the same heterogeneous space scenario and study the relationship between predator and

prey geographic distributions. The derivation of the model builds on the one-species model described in M´endez-Vera et al. 2019. It follows that of Case and Taper 2000, but here we propose a predator-prey scenario instead of a competition one (refer to the Appendix 3.4.1 for details). We consider two interacting prey and predator populations distributed along onedimensional space parametrized by the variable −1 < x < 1, with dynamics taking place over continuous time t 0. We suppose populations are structured by a phenotypic variable −1 < z < 1, and that there is an optimal phenotypic value for each spatial position, which we will denote θ1(x) = b1x for prey and θ2(x) = b2x for predators. Note that because b1 and b2 may differ, the environmental cline may be relatively harsher for one of the two species.

**Table of contents :**

ACKNOWLEDGEMENTS

LIST OF FIGURES

ABSTRACT

R´ESUM´E

**CHAPTER I. Introduction **

1.1 A few words on current challenges in understanding species distributions

1.2 Facing the challenges: predicting and managing changes in species distributions

1.2.1 The heterogeneity of environmental variables as a spatial determinant of species distributions

1.2.2 Dispersal processes

1.3 Effects of evolution on spatial distributions

1.4 Effects of interactions on spatial distributions

1.5 Theoretical approaches to spatial ecology

1.5.1 Spatially implicit models

1.5.2 Spatially explicit models without evolution

1.5.3 Spatially explicit models with adaptation

1.6 Organization and general idea of this thesis

1.7 Work Summary

1.7.1 Evolutionary invasion speeds and invasion mechanisms

1.7.2 Effects of predation on evolutionary invasion speeds and species distributions

1.7.3 Pathogen-aided invasions and adaptation to pathogens

**II. Evolutionary invasion speeds and invasion mechanisms **

2.1 Introduction

2.2 Kirkpatrick and Barton’s model for a single species’ range evolution along a linear gradient

2.3 Explicit approximation of propagation speeds under various adaptation scenarios

2.4 Relating propagation speeds to adaptation regimes

2.5 Discussion

2.6 Appendices

2.6.1 Infinite adaptation case

2.6.2 Numerical schemes

**III. Effects of predation on evolutionary invasion speeds and species distributions **

3.1 Introduction

3.2 Effects of dispersal and adaptation potential on the evolution of species ranges in a predator-prey framework

3.2.1 Model presentation and main assumptions

3.2.2 Homogeneous equilibria and coexistence

3.2.3 Propagation fronts and intrinsic propagation speeds

3.2.4 Front classification and geographic distribution

3.3 Discussion

3.4 Appendices

3.4.1 Model derivation

3.4.2 Numerical schemes

3.4.3 Proof of expression (3.3)

**IV. Retracting fronts in pathogen-aided invasions **

4.1 Introduction

4.2 Model presentation

4.2.1 Eco-evolutionary dynamics in time and space

4.2.2 Ecological considerations and constraints

4.2.3 Evolutionary approximations ignoring spatial structure

4.3 Adaptation in the spatial model

4.4 Results

4.4.1 First set of parameters

4.4.2 Second set of parameters

4.4.3 Summary of the results

4.5 Discussion

4.6 Appendix

4.6.1 Model derivation

4.6.2 What is an “adapted trait”?

**V. Synthesis and Discussion **

5.1 Understanding the effects of evolution through propagation speeds

5.2 Under what conditions can we make predictions about the species distributions?

5.3 Possible applications

5.4 What kind of extensions are possible?

5.5 Final thoughts and perspectives

**BIBLIOGRAPHY **