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## Evolutionary invasion speeds and invasion mechanisms

Current global changes make it urgent to understand the eco-evolutionary deter-minants 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 monospe-cific 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: ∂n = ∂2n + n 1 − n − 1 (¯z − Bx)2 , (1.7a) ∂t ∂x2 2.

**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 under-standing 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 pre-dation 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 interac-tions and adaptation.

Here, we propose a spatially explicit model accounting for predator-prey interac-tions and adaptation to environment along an optimal phenotypic cline.

We consider two interacting prey and predator populations distributed along one-dimensional space, parameterized by the variable −∞ < x < ∞, with dynamics taking place over time t ≥ 0. We suppose that populations are structured by a phenotypic variable −∞ < z < ∞ and that, for each population, there is an optimal phenotypic trait value θ1(x) = b1x (for prey) and θ2(x) = b2x (for predators).

Denoting by n1(t, x) and n2(t, x) the local density of prey and predators, respec-tively, and by z¯1(t, x) and z¯2(t, x) their respective mean trait value, then the dynamics of the system are given by: ∂n1 ∂2n1 1 2 = + n1 1 − n1 − (b1x − z¯1) − βn2.

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

The one-species model proposed by Kirkpatrick and Barton 1997 is a spatially explicit model in heterogeneous space accounting simultaneously for migration eﬀects and adaptation. It assumes individuals are characterized by a phenotypic trait and that heterogeneity in space is given by a continuous cline of the optimal value for this phenotype. Individuals whose phenotype deviates from this optimum will suﬀer a fitness penalty. Although this model assumes that the environmental cline remains fixed in time, it provides a framework to study, for example, the eﬀects of climate change on species distributions (see e.g. Norberg et al. 2012), as it can easily be modified to include a time-varying environment. It is also suitable to study invasion scenarios, linking the characteristics of the environment and those of the introduced population.

The model assumes infinite one-dimensional linear space and considers local pop-ulation density n(t, x), i.e. the density of individuals at location x at time t ≥ 0, and the mean phenotypic value of this population, z¯(t, x), at this time and this lo-cation. The environmental cline is modeled through the optimal phenotype function θ(x) = Bx meaning the optimal value varies linearly through space. After a renormal-ization of the original variables and parameters in the full system (see Kirkpatrick and Barton 1997 for details), the equations governing density and phenotype dynamics are given by:

∂n = ∂2n + n 1 − n − 1 (¯z − Bx)2 , (2.1a).

∂z¯ = ∂2z¯ + 2 ∂ log n ∂z¯ − A (¯z − Bx) , (2.1b).

where A is a measure of adaptation potential of the species (A is proportional to genetic variance) and B is the rate of change of the optimal phenotype through space, also considered to be a measure of spatial heterogeneity.

System (2.1) describes the eco-evolutionary dynamics of the species under local adaptation and spatial diﬀusion. The first term of Equation (2.1a) models the disper-sal of the population through a diﬀusion process. The second term contains the local ecological dynamics, corresponding to the logistic model and a penalizing term that captures local maladaptation. The first term of equation (2.1b) models the diﬀusion of genes that is linked to the diﬀusion of individuals, while the second term corrects for asymetries in gene flows (gene flow being more important from large populations to small populations than the other way round). The third term corresponds to the eﬀects of local adaptation due to directional selection, driving the mean phenotype value z¯ toward the local optimum Bx at a rate A.

Migration and adaptation potential A have antagonistic eﬀects, whose results vary depending on the spatial heterogeneity B. Depending on A and B, the population may survive in a limited space (for intermediate values of A and B), may invade the whole space (when adaptation is larger than a certain critical value, allowing the population to surmount spatial heterogeneity) or may become extinct (when adap-tation is too small with respect to spatial heterogeneity; (Kirkpatrick and Barton 1997)). This result can be partially re-stated in terms of propagation speeds (Fisher 1937), which answer at the same time the question of geographic dynamics of the population: if we consider as initial condition a geographic frontier, i.e., the initial condition is n(0, x) = 1 for x ≤ 0 and 0 otherwise, with the species being perfectly locally adapted (¯z(0, x) = Bx) wherever it is present (n(0, x) = 1), then the solutions behave like propagating fronts with a characteristic speed. For Kirkpatrick and Bar-ton’s one-species model, the direction and magnitude of the advancing front depend on the parameters A and B. Positive speeds mean the front moves towards positive values of x so that the species progressively invade (hereafter invasion fronts). On the contrary, negative values mean that the species distribution retracts (either to a limited range or toward the extinction of the species, hereafter extinction fronts). We dub cKB (A, B) the speed of the solution of system (2.1) for parameters A and B.

In terms of propagation speeds, species whose borders correspond to invasion fronts are able to continuously adapt to new environments and thus will always be able to invade the whole space. On the contrary, negative speed fronts only mean maladapted gene flow is stronger than adaptation, causing local extinctions that can lead to two outcomes: either the population becomes extinct, or two fronts from diﬀerent directions collide canceling out maladaptations in the center and allowing the species to survive in a limited space. We cannot distinguish between these two last outcomes based on speed alone, another demographic criterion is needed to do so. Refer to Figure 2.1 for a clearer link between propagation speeds and spatial distribution.

#### Explicit approximation of propagation speeds under various adaptation scenarios

We investigate the variation of propagation speeds for diﬀerent values of parame-ters A and B, with a focus on the two limit cases of infinitely strong adaptation and very weak adaptation potentials. Even though it is unlikely species adapt infinitely fast, the variation in propagation speeds between these two limit cases can tell us when a finite adaptation is strong enough so that it is qualitatively infinite.

One first important result is that when adaptation goes to infinity, A → ∞, the system (2.1) becomes the Fisher-KPP equation (after Fisher 1937 and Kolmogorov et al. 1937, see the Appendix 2.6.1 for details), given by ∂n = ∂2n + n (1 − n) (2.2).

in its non-dimensional form (refer to the appendix for details on this infinite adaptation limit). Its solutions are traveling fronts with a minimal admissible speed of √ cF = 2 (or, in its dimensional form, c∗F = 2 rδ with r corresponding to the intrinsic growth rate of the population and δ a measure of its dispersal), so that for infinite adaptation potential invasion speed is finite and constant. Equation (2.2) has an infinity of solutions for diﬀerent front speeds c ≥ cF , but cF is the smallest one and the only one with biological meaning.

We can draw two other important conclusions thanks to equation (2.2). First, in an ecological context, the Fisher-KPP equation can only model propagation of species whose adaptation is so fast that they are continuously well-adapted everywhere, since the equation is the same as the system (2.1) neglecting maladaptation (and all the terms involving the phenotypic trait). Second, invasion speeds for the one-species model given by system (2.1) will always be lower than cF = 2, since growth rate in the Fisher-KPP model is always larger than the one of the KB model, because maladaptation eﬀects can only decrease population fitness (having thus a negative eﬀect on speed). This means that the maximum speed of range expansion is only √ constrained by the species growth rate and dispersal ability (since it is c∗F = 2 rδ in its dimensional form).

**Table of contents :**

**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 structure132

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**