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1701.07861v2_0

http://arxiv.org/abs/1701.07861v2

Δ δ̣ ρ̊ π α γ → σ β̱ ϵ Γ ω łλ ϕ ψ μ τ χ̧1cm Diversity and coevolutionary dynamics in high-dimensional phenotype spaces Michael Doebeli^∗ & Iaroslav Ispolatov^∗∗^∗ Departments of Zoology and Mathematics, University of British Columbia,6270 University Boulevard, Vancouver B.C. Canada, V6T 1Z4; doebeli@zoology.ubc.ca ^∗∗ Departamento de Fisica, Universidad de Santiago de ChileCasilla 302, Correo 2, Santiago, Chile; jaros007@gmail.com December 30, 2023

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Chile; jaros007@gmail.com December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================10mm 10 mm Supporting Material: Appendix A 10 mm RH: Diversity and coevolutionary dynamics 10 mm Corresponding author:

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dynamics 10 mm Corresponding author: Michael Doebeli, Department of Zoology and Department of Mathematics,University of British Columbia, 6270 University Boulevard, Vancouver B.C. Canada, V6T 1Z4, Email: doebeli@zoology.ubc.ca.Abstract 0.3cm We study macroevolutionary dynamics by extending microevolutionary competition models to long time scales. It has been shown that for a general class of competition models, gradual evolutionary change in continuous phenotypes (evolutionary dynamics) can be

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phenotypes (evolutionary dynamics) can be non-stationary and even chaotic when the dimension of the phenotype space in which the evolutionary dynamics unfold is high. It has also been shown that evolutionary diversification can occur along non-equilibrium trajectories in phenotype space.We combine these lines of thinking by studying long-term coevolutionary dynamics of emerging lineages in multi-dimensional phenotype spaces. We use a statistical approach to investigate the evolutionary dynamics

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approach to investigate the evolutionary dynamics of many different systems. We find: 1) for a given dimension of phenotype space, the coevolutionary dynamics tends to be fast and non-stationary for an intermediate number of coexisting lineages, but tends to stabilize as the evolving communities reach a saturation level of diversity; and 2) the amount of diversity at the saturation level increases rapidly (exponentially) with the dimension of phenotype space.These results have implications for

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space.These results have implications for theoretical perspectives on major macroevolutionary patterns such as adaptive radiation, long-term temporal patterns of phenotypic changes, and the evolution of diversity. 1 cm Keywords: Long-term evolution | Diversity and stability | Adaptive radiation § INTRODUCTIONOne of the fundamental problems in evolutionary biology is to understand how microevolutionary processes generate macroevolutionary patterns. In particular, the emergence of

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patterns. In particular, the emergence of macroevolutionary changes in the speed of evolution <cit.>, and of macroevolutionary changes in patterns of species diversity <cit.> have long been of great interest. For example, <cit.> have recently proposed that over macroevolutionary time scales, relatively short intermittent bursts of high rates of evolutionary change should alternate with long periods of bounded phenotypic fluctuations.Also, there is much discussion about whether species diversity

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much discussion about whether species diversity saturates over evolutionary time in a given environment <cit.>. Phylogenetic analysis has been used to shed light on these questions <cit.>, but mechanistic models in which short-term ecological interactions are extrapolated to yield long-term patterns of diversity and evolutionary change have only recently been developed. Most of these models have been used to study the long-term evolution of diversity by analyzing processes of community assembly

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by analyzing processes of community assembly emerging from short-term ecological dynamics <cit.>. In particular, these papers have mainly focussed on how diversity changes over time, but not on how the nature of the coevolutionary dynamics of a given set of coexisting species changes as the diversity changes. In fact, in all these models, the evolutionary dynamics for a fixed amount of diversity, i.e., for a given set of species, converge to an equilibrium. However, if one wants to understand

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equilibrium. However, if one wants to understand macroevolutionary changes in the “tempo and mode” <cit.> of evolution, one not only needs to consider how diversity changes over evolutionary time, but also how such changes in diversity affect the nature of evolutionary dynamics <cit.>. Indeed, there is evidence from evolution experiments with microbes that evolutionary dynamics in more diverse communities are qualitatively different from the evolutionary dynamics in less diverse communities

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evolutionary dynamics in less diverse communities <cit.>. Here we present a theoretical investigation of the questions of how diversity affects the complexity of coevolutionary dynamics.In general, the number of different phenotypes that affect ecological and evolutionary processes is an important quantity. For example,determining the dimensionality of niche space in ecological food webs is a classical problem <cit.>, and it has recently been shown that including more phenotypic dimensions in

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that including more phenotypic dimensions in models for community assembly has a strong effect on the structure of the emerging food webs <cit.>. Implicitly, the importance of the dimension of phenotype space is also acknowledged in phylogenetic research through the notion of “adaptive zones” <cit.>. In particular, it is thought that much of the extant diversity has evolved as a consequence of lineages entering new adaptive zones, which can be interpreted from the phenotypic perspective as an

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interpreted from the phenotypic perspective as an increase in the dimension of phenotype space. In general, given the large number of phenotypic properties that determine an individual's life history and ecology in almost any species, one would expect that ecological interactions are generally determined by many phenotypic properties, and that selection pressures emerging from ecological interactions in turn affect many phenotypes simultaneously. For example, comprehensive modelling of the

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For example, comprehensive modelling of the metabolic network in E. coli cells comprises more than 2000 reactions <cit.>. These reactions are in turn controlled by thousands of genes in a complicated interaction network whose exact workings are largely unknown. Nevertheless, many of the genes contributing to this network of metabolic reactions will be under selection in any given environmental setting, and as a consequence, a large number of phenotypic properties have the potential to undergo

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properties have the potential to undergo evolutionary change. It is generally not known how exactly these phenotypic properties impinge on birth and death rates of individual organisms, and hence what exactly the ecological selection pressures are on these properties. Nevertheless, it seems clear that in general, many phenotypes will evolve at the same time, i.e., that evolution generally takes place in high-dimensional phenotype spaces.We have recently argued that if evolution takes place in

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recently argued that if evolution takes place in high-dimensional phenotype spaces, then the evolutionary dynamics, that is, the phenotypic change over evolutionary time, can be very complicated, i.e., non-stationary and often chaotic <cit.>. In low-dimensional phenotype spaces, non-equilibrium evolutionary dynamics are less likely. However, if a species evolving on a simple attractor gives rise to diversification, the effectivedimensionality of the evolving system increases, as the species

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of the evolving system increases, as the species that emerge from diversification coevolve, driven by both intra- and interspecific ecological interactions.Thus the total dimensionality of the resulting dynamical system describing multispecies coevolution is the number of species times the dimensionality of the phenotype space in which each species evolves. Based on our earlier results <cit.>, one could then expect that due to the increase in dimensionality, diversificationleads to more

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in dimensionality, diversificationleads to more complicated evolutionary dynamics in each of the coevolving species. On the other hand, as a multispecies community becomes more diverse and evolves towards saturation, the available niches tend to get filled, and hence evolutionary change has to become highly coordinated between interacting species and thus constrained, potentially leading to simplified evolutionary dynamics. It is thus unclear how the nature of the evolutionary dynamics changes

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the nature of the evolutionary dynamics changes as the pattern of diversity changes during community assembly.We investigate these issues by applying the framework of adaptive dynamics <cit.>to a general class of competition models. The main question we address is, how does the complexity of long-term coevolutionary dynamics depend on the diversity of the coevolving community? We show that in low-dimensional phenotype spaces, there is a humped-shaped relationship between diversity and the

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relationship between diversity and the complexity of evolutionary dynamics: in communities with low diversity, coevolutionary dynamics are often simple, i.e., stationary in the long-time limit; for intermediate degrees of diversity, non-stationary (complex) coevolutionary dynamics are common, and each of the species in the community evolves on a complicated trajectory in phenotype space; and for high amounts of diversity, coevolutionary dynamics become simple again, i.e., stationary. In

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become simple again, i.e., stationary. In particular, as communities reach diversity saturation, e.g. through adaptive diversification <cit.>, coevolutionary dynamics change from complex to simple. Our results are relevant for a number of issues concerning patterns of macroevolution. For example, the results suggest that during processes of adaptive radiation <cit.>, evolutionary dynamics are more complicated early in the radiation than late in the radiation, a pattern that corresponds to the

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the radiation, a pattern that corresponds to the “early-burst” perspective of macroevolution that has attracted much attention in recent years <cit.>. Our results also show that the level at which diversity saturates depends on the dimensionality of phenotype space, with higher dimensions allowing for more diversity. This observation is in accordance with data from radiations in fishes <cit.> and points to the possibility of a microevolutionary mechanism for the “blunderbass theory" of temporal

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for the “blunderbass theory" of temporal patterns of macroevolutionary changes and diversification <cit.>: if evolution operates on the dimension of phenotype space on a very slow time scale, then on shorter time scales diversity may saturate and thereby generate relatively stationary evolutionary dynamics, whereas on longer time scales the dimension of phenotype space may increase, e.g. due to gene duplications, thus generating a new burst of non-equilibrium (co-)evolutionary dynamics until

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non-equilibrium (co-)evolutionary dynamics until the diversity reaches a new saturation level. Such patterns of intermittent bursts have recently been found in the phylogenies of birds and echinoids <cit.>, and the bursts have been attributed to the evolution of flight capabilities and of novel feeding techniques, respectively, both of which can be interpreted as an increase in the dimensionality of the relevant phenotype space. This perspective may also shed light on the question of whether

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may also shed light on the question of whether diversity saturates or not <cit.>: diversity may saturate for a given dimension of phenotype space, but evolutionary innovation in the form of new phenotypic dimensions may intermittently generate room for additional bouts of evolutionary diversification. § METHODS §.§ Single-cluster adaptive dynamicsAs in <cit.>, we study a general class of models for frequency-dependent competition in which ecological interactions are determined by d-dimensional

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interactions are determined by d-dimensional phenotypes, where d≥1.For simplicity, we consider homogeneous systems, so no spatial coordinates are included.The ecological interactions are described by a competition kernel α(𝐱, 𝐲) and by a carrying capacity K(𝐱), where 𝐱,𝐲∈ℝ^d are the d-dimensional continuous phenotypes of competing individuals. The competition kernel αmeasures thecompetitive impact that an individual of phenotype 𝐱 has on an individual of phenotype 𝐲, and we assume thatα(𝐱, 𝐱)=1

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of phenotype 𝐲, and we assume thatα(𝐱, 𝐱)=1 for all 𝐱. Assuming logistic ecological dynamics, K(𝐱) is then the equilibrium density of a population that is monomorphic for phenotype 𝐱. The adaptive dynamics of the phenotype 𝐱 is a system of differential equations for d𝐱/dt. To derive the adaptive dynamics, one defines the invasion fitness f(𝐱, 𝐲) as the per capita growth rate of a rare mutant phenotype 𝐲 in the monomorphic resident 𝐱 population that is at its ecological equilibrium K(𝐱):f(𝐱, 𝐲)

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is at its ecological equilibrium K(𝐱):f(𝐱, 𝐲) = 1 - α(𝐱, 𝐲) K(𝐱)/K(𝐲). The expression for the invasion fitness reflects the fact that the growth rate of the mutant 𝐲 is negatively affected by the effective density experienced by the mutant, α(𝐱, 𝐲) K(𝐱), discounted by the carrying capacity K(𝐲) of the mutant (see <cit.> for more details).Note that f(𝐱, 𝐱)=0 for all 𝐱. The invasion fitness f(𝐱, 𝐲) gives rise to the selection gradients in the i=1,...,d phenotypic components: s_i(𝐱) ≡∂ f(𝐱, 𝐲)/∂

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phenotypic components: s_i(𝐱) ≡∂ f(𝐱, 𝐲)/∂ y_i |_𝐲=𝐱 =- ∂α(𝐱, 𝐲)/∂ y_i |_𝐲=𝐱 + ∂ K(𝐱)/∂ x_i1/K(𝐱), The selection gradients in turn define the adaptive dynamics as a system of differential equations on phenotype space ℝ^d, which is given byd𝐱/dt = 𝐌(𝐱)·𝐬(𝐱).Here 𝐬(𝐱) is the column vector (s_1(𝐱),...,s_d(𝐱)), and 𝐌(𝐱) is the mutational variance-covariance matrix. In this matrix, the diagonal elements contain information about the size and rate of mutations in each of the phenotypic dimensions,

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mutations in each of the phenotypic dimensions, whereas the off-diagonal elements contain information about the covariance between mutations in two different phenotypic dimensions. This matrix essentially captures “evolvability” of a population andgenerally depends on the current resident phenotype 𝐱, and influences the speed and direction of evolution. For simplicity, we assume here that this matrix is the identity matrix. For more details on the derivation of the adaptive dynamics (<ref>) we

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derivation of the adaptive dynamics (<ref>) we refer to a large body of primary literature (e.g. <cit.>). We note that the adaptive dynamics (<ref>) can be derived analytically as a large-population limitof an underlying stochastic, individual-based model that is again defined based on the competition kernel α(𝐱, 𝐲) and the carrying capacity K(𝐱) <cit.>. Specifically, here we consider a class of systems that are defined by competition kernels of the form(𝐱,𝐲)=exp [∑_i,j=1^d b_ij(x_i-y_i)x_j

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of the form(𝐱,𝐲)=exp [∑_i,j=1^d b_ij(x_i-y_i)x_j -∑_i=1^d(x_i-y_i)^2/2_i^2]. Here the coefficients b_ij in the first sum on the right hand side are arbitrary and correspond to the simplest form of a generic, non-symmetric competition kernel that can generate non-stationary evolutionary dynamics. It can be interpreted as the lowest-order (non-trivial) term from a Taylor expansion of an unknown non-symmetric competition function. Adaptive dynamics of asymmetric competition has been studied quite

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of asymmetric competition has been studied quite extensively (e.g. <cit.>), and is necessary to generate single-species non-equilibrium dynamics in high-dimensional phenotype spaces <cit.>. The second sum on the right hand side represents “Gaussian competition”, according to which the competitive impact between individuals increases with phenotypic similarity between the competing individuals. The parameters _i measure how fast the effect of competition declines as phenotypic distance in the

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declines as phenotypic distance in the i-component increases. For the carrying capacity we assume K(𝐱)=exp(-∑_i^d x_i^4/4). This implies that the carrying capacity imposes an element of stabilizing selection for the phenotype 𝐱=0, at which the carrying capacity is maximal. Thus, the frequency-dependent component of selection is generated by the competition kernel, whereas the frequency-independent component of selection is due to the carrying capacity. With these assumptions, the adaptive

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capacity. With these assumptions, the adaptive dynamics (<ref>) become d x_i/dt=∑_i=1^d b_ijx_j - x_i^3,i=1,...,d. We note thatthe terms -x_i^3 in (<ref>) are due to the carrying capacity and serve to contain the trajectories of (<ref>) in a bounded domain of phenotype space. Also, the Gaussian part of the competition kernel does not affect the adaptive dynamics of monomorphic populations, i.e., the _i do not appear in (<ref>), because the Gaussian part always has a maximum at the current

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Gaussian part always has a maximum at the current resident, and hence the corresponding first derivative in the selection gradient (<ref>) is 0. The system of ODEs(<ref>) describes the trajectory of an evolving monomorphic population in phenotype space ℝ^d. In <cit.> we have shown that for general competition kernelssuch trajectories can be very complicated, particularly when the dimension d is large. With complex evolutionary dynamics, trajectories can be quasi-periodic or chaotic, and

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can be quasi-periodic or chaotic, and typically visit many different regions of phenotype space over evolutionary time. When d is low the dynamics tend to be simpler, and often converge to an equilibrium attractor.We can assess the likelihood of equilibrium dynamics for a given dimension d by choosing the d^2 coefficientsb_ij in (<ref>) randomly and independently, e.g. from a normal distribution with mean 0 and variance 1, solving the resulting adaptive dynamics (<ref>) and checking whether it

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adaptive dynamics (<ref>) and checking whether it converges to an equilibrium. If this is done repeatedly, we can approximate the probability of equilibrium dynamics as the fraction of runs that converged to an equilibrium. For d=1 the probability of equilibrium dynamics is of course 1, and for d=2,3,4, the resulting probabilities of equilibrium dynamics are approximately 85%, 81% and 74%, respectively. These are the dimensions that we will primarily use in the analysis presented below, but we

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use in the analysis presented below, but we note that the probability of equilibrium dynamics goes to 0 for large d <cit.>. §.§ Multi-cluster adaptive dynamicsHere we are interested in the question of how diversification and subsequent coexistence of species (also called phenotypic clusters or simply clusters through the text) affects the evolutionary dynamics. While the Gaussian term in the competition kernel (<ref>) does not affect the adaptive dynamics of single monomorphic populations, this

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dynamics of single monomorphic populations, this term is crucial for determining whether evolutionary diversification occurs. For one-dimensional phenotype spaces (d=1) this is very well known and is encapsulated in the concept of evolutionary branching <cit.>. An evolutionary branching point is an equilibrium point of (<ref>) that is both an attractor for the adaptive dynamics and a fitness minimum. The reason that such points exist in the competition models considered here is precisely that

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models considered here is precisely that the Gaussian term does not affect the adaptive dynamics, but does affect the curvature of the fitness landscape, i.e., the second derivative of the invasion fitness (<ref>). In particular, small enough _i's in the Gaussian term will make any equilibrium point a fitness minimum, and hence will give rise to evolutionary diversification. Evolutionary branching in scalar traits has been described in a plethora of different models (for an overview we refer to

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of different models (for an overview we refer to Eva Kisdi's website at the Department of Mathematics and Statistics at the University of Helsinki, http://www.mv.helsinki.fi/home/kisdi/addyn.htm). In high-dimensional phenotype spaces, equilibrium points of (<ref>) can also be fitness minima along some directions in phenotype space. For this to happen the Hessian matrix of second derivatives of the invasion fitness (<ref>), evaluated at the equilibrium, must have positive eigenvalues. Indeed, in

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must have positive eigenvalues. Indeed, in higher dimensional phenotype spaces the conditions for the existence of positive eigenvalues of this Hessian matrix, and hence for diversification, generally become less stringent <cit.>. Importantly, evolutionary diversification can also occur from non-equilibrium adaptive dynamics trajectories <cit.>. If the adaptive dynamics (<ref>) exhibit non-equilibrium dynamics, the crucial quantity determining whether diversification occurs is again the Hessian

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diversification occurs is again the Hessian matrix of second derivatives of the invasion fitness (<ref>), but now restricted to the subspace of phenotype space that is orthogonal to the selection gradient <cit.>. Essentially, diversification can occur in orthogonal directions in which this Hessian has positive curvature, and hence in which the invasion fitness has a minimum. Because the population is still evolving along the selection gradient, elucidating the exact conditions for

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gradient, elucidating the exact conditions for diversification requires a careful analysis <cit.>. In the present context, the implication of these results is that, just as with equilibrium adaptive dynamics, diversification can occur along non-equilibrium trajectories of (<ref>) if the _i in the Gaussian term of the competition kernel are small enough, i.e., if the frequency dependence generated by Gaussian competition is strong enough <cit.>. To investigate the process of diversification and

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To investigate the process of diversification and the subsequent coevolutionary dynamics, we extend the adaptive dynamics (<ref>) to several coexisting phenotypic clusters as follows. We assume that an evolving community consists of m monomorphic populations, each given by a phenotype 𝐱_r, r=1,...,m, with phenotypic components x_ri, i=1,...d (where d is the dimension of phenotype space). Let N_r be the population density of cluster 𝐱_r. Then the ecological dynamics of the m clusters are given

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ecological dynamics of the m clusters are given by the system of logistic differential equations d N_r(t)/ d t = N_r( t)( 1 - ∑_s=1^m (𝐱_s, 𝐱_r) N_s ( t)/K(𝐱_r)),r=1,...,m. Let N_r^*, r=1,...,m denote the equilibrium of system (<ref>) (more generally, for the purposes of deriving the adaptive dynamics, the quantities N_r^* are suitable time averages of population densities over the ecological attractor of (<ref>); however, our extensive numerical simulations indicated that (<ref>) always

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simulations indicated that (<ref>) always converges to an equilibrium). Making the traditional adaptive dynamics assumption that ecological dynamics occur on a faster time scale than evolutionary dynamics, we calculate the invasion fitness function in cluster r based on the densities N_r^* of the various clusters:f(𝐱_1,...,𝐱_m,𝐱_r')=1 - ∑_s=1^m (𝐱_s,𝐱_r')N_s^*/K(𝐱_r'). Here 𝐱_1,...,𝐱_m describe the phenotypic state of the resident population, and 𝐱_r' denotes the mutant trait in cluster r,

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and 𝐱_r' denotes the mutant trait in cluster r, r=1,...,m. Taking the derivative of (<ref>) with respect to 𝐱_r' and evaluating it at the resident, 𝐱_r'=𝐱_r, yields the components of the selection gradient 𝐬_r for the cluster r as: s_ri= ∑_sN_s^*(-1/K(𝐱_r).∂(𝐱_s,𝐱_r')/∂x_ri'|_𝐱_r'=𝐱_r +(𝐱_s,𝐱_r)/K^2(𝐱_r)∂K(𝐱_r)/∂ x_ri),i=1,...,d. For coevolutionary adaptive dynamics, one has to take into account that the rate of mutations in each evolving phenotypic cluster is proportional to the current

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phenotypic cluster is proportional to the current population size of that cluster <cit.>, and hence that the speed of evolution is influenced by the population size. In the single-cluster system such consideration only rescales time without affecting the geometry of the trajectory and thus is usually ignored. However, in the multi-cluster system, instead of assuming that the mutational process is described by the identity matrix as in (<ref>), we now assume that in each cluster r, the

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we now assume that in each cluster r, the mutational variance-covariance matrix M_r is a diagonal matrix with entries N_r^*. This generates the following d· m differential equations describing the adaptive dynamics in the coevolving community: d x_ri/dt= N_r^* s_ri, i=1,…,d, r=1,…, m. For the multicluster adaptive dynamics, the equation (<ref>,<ref>,<ref>) replacetheir single-cluster analogs (<ref>,<ref>,<ref>). It is important to note that the Gaussian part of the competition kernelnot only

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Gaussian part of the competition kernelnot only affects whether diversification occurs, but in contrast to the adaptive dynamics of single monomorphic populations, the Gaussian term will indeed affect the coevolutionary adaptive dynamics (<ref>) of the phenotypic clusters that coexist after diversification has occurred, because it affects both the ecological dynamics (<ref>) and the selection gradient (<ref>). §.§ Numerical procedureTo study diversification and subsequent multi-cluster adaptive

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and subsequent multi-cluster adaptive dynamics, we implemented the following iterative numerical scenario:0.5 cm Step 1: Each simulation run is initiated with a randomly generatedd × d matrix of the coefficientsb_ij forthe competition kernel (<ref>). The coefficients are drawn from a Gaussian distribution with zero mean and d^-1/2 variance. As explained in <cit.>, this is done to keep the sum of the d terms ∑_j=1^d b_ij x_j in (<ref>) of orderx_i, i.e. independent of d. Then a certain number of

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i.e. independent of d. Then a certain number of clusters, given by a parameter m_0, each with population size of order 1, are randomly placednear the phenotype 0, i.e., near the maximum of the carrying capacity. 0.5 cm Step 2: For a given set of phenotypic clusters, the population dynamics of all clusters is solved using the ecological dynamics (<ref>). The system of differential equations is integrated using a 4th-order Runge-Kutta algorithm for ∼ 10^3 time steps of duration dt ∼ 10^-2 to

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for ∼ 10^3 time steps of duration dt ∼ 10^-2 to ensure convergence to the equilibrium (or, in case there is no such convergence, to ensure a correct calculation of the time average of the various population densities). If the population density of a givencluster falls below the threshold N_min∼10^-8, the cluster is eliminated from the system. During the ecological dynamics the evolutionary dynamics is frozen and evolutionary time does not advance.0.5 cm Step 3: After calculating the N_r^*,

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cm Step 3: After calculating the N_r^*, r=1,...,m (where m is the current number of clusters), the adaptive dynamics of the phenotypes of the clusters is advanced via (<ref>,<ref>) using a 4th-order Runge-Kutta algorithm with a typical time-step d∼ 10^-2, by which the evolutionary time is advanced as well. After this evolutionary time step, the ecological dynamics are recalculated, potentially preceded by the following step 4, which is only performed if the corresponding time condition is

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performed if the corresponding time condition is satisfied. 0.5 cm Step 4: The level of diversity, i.e., the number of clusters in the system, is controlled as follows. Each _c time units the distances between clusters are assessed. If the distance between two or more clusters is below a thresholdx ∼ 10^-3, these clusters are merged, preserving the total population size of the merged clusters and the position of their centre of mass.Immediately after this comparison step, the total number of

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after this comparison step, the total number of clusters is compared to the target number of clusters, which is given by a system parameter m_max. If the current number of clusters is below m_max, a new cluster is created by randomly picking an existing cluster,splitting it in half and separating the two new clusters in a random direction in phenotype space bythe distanceof the merging threshold, x. 0.5 cm Step 5: In our simulations, we take measurements at regular time intervals (ranging from

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at regular time intervals (ranging from _m∼ 1 - 10 time units). One of the main quantities of interest is the average per capita evolutionary speed v in the evolving community,which is the average of the norms of the vectors of trait variation (evolution) rates in each cluster, weighted by the cluster population size,computed as v=∑_r=1^mN_r√(∑_i=1^d (d x_ri/dt)^2)/∑_r=1^m N_rThis quantity is a strong indicator of the nature of the evolutionary dynamics of the coevolving system. In particular,

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dynamics of the coevolving system. In particular, our very extensive numerical simulations indicate that when the average speed falls below 10^-2, then the system eventually exhibits equilibrium evolutionary dynamics. In contrast, when the average evolutionary speed remains high, the coevolving system tends to exhibit complicated, non-equilibrium dynamics, with the majority of the clusters exhibiting large fluctuations in phenotype space over evolutionary time. An example of such

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space over evolutionary time. An example of such non-equilibrium coevolution is given in the next section. Other measurements include the position and population size of all clusters in the system,and the number of “distinct” clusters separated by a “visible” distance X=0.1. These measurements can also be averaged over time. 1cm For any given simulation run initiated by step 1 above, steps 2-5 were repeated iteratively until a specified final simulation time is reached, or until evolution comes

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time is reached, or until evolution comes to a halt, which by our definition occurs when the average evolutionary speed falls below a threshold, v<10^-4.Our general approach consisted of simulating many different systems according to the above scheme, and then computing statistical characteristics such as the fraction of runs that result in non-equilibrium dynamics, or the average evolutionary speed as a function of the level of diversity (see Results section). One crucial feature of our

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(see Results section). One crucial feature of our algorithm is the periodic generation of new clusters in step 4, which mimics diversification events, i.e., evolutionary branching. Diversification is thus modeled by simply adding new phenotypic clusters at certain points in time and close to existing clusters. This mimics the sympatric split of an ancestral lineage. Sympatric diversification is a theoretically robust phenomenon <cit.> and our procedure represents a shortcut for this phenomenon

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represents a shortcut for this phenomenon necessitated by computational feasibility. If such splitting is not feasible given the current ecological circumstance, the new cluster will not diverge phenotypically from the ancestor, and hence will be merged again with the ancestor (see below). Alternatively, newly generated clusters may go extinct ecologically. In either case, speciation was not successful. Thus, in our models it is the ecological circumstances that determine whether speciation can

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that determine whether speciation can occur or not, but the process of speciation itself (i.e., the splitting) is performed in a simplified manner. If speciation is successful and the newly generated clusters diverge and persist ecologically, then diversity has increased (unless other clusters go extinct). We note that by construction, the maximal level of diversity in a given simulation run, i.e., the number of different clusters, cannot exceed the parameter m_max. Therefore, this parameter

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the parameter m_max. Therefore, this parameter allows us to control the level of diversity in a given simulation.There are in principle other, less artificial ways to model diversification. In particular, stochastic, individual-based based models and partial differential equation models <cit.> have been used to describe the evolutionary dynamics of phenotype distributions. In such models, diversification is an emergent property that is reflected in the formation of new modes in the evolving

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in the formation of new modes in the evolving phenotype distributions. While these techniques are very useful in general, they are currently not computationally feasible for the statistical approach that we employed here, which requires systematic simulation of many different systems. Also, they would not allow for control of the level of diversity, as the number of phenotypic modes would simply be an emergent property of the evolving system. Nevertheless, we have used these alternative

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Nevertheless, we have used these alternative techniques to illustrate the robustness of salient results using particular examples. A more detailed description of these techniques is given in theAppendix. Another alternative would be to assume that new clusters (species) are assigned phenotypes that are chosen randomly in phenotype space, rather than close to an existing cluster. This could correspond to immigration of new species into an existing community. However, we would not expect this to

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community. However, we would not expect this to affect our main results, because with complicated evolutionary dynamics, the initial phenotypic position of a given cluster becomes irrelevant after some time. Finally, we note that the merging of clusters (species) is done solely for computational reasons and has no biological meaning (apart from designating organisms that are closely related and phenotypically very close as belonging to the same species). Merging of clusters only occurs shortly

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species). Merging of clusters only occurs shortly after a new cluster is seeded close to an existing one, and only if the new cluster does not diverge from the existing one (i.e., only if the ecological conditions for diversification are not satisfied). If divergence is successful, the clusters will never again get close enough to other clusters to be merged because of the repelling force of frequency-dependent competition. Thus, the only function of merging is to prevent the number of clusters

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of merging is to prevent the number of clusters from artificially becoming very large.§ RESULTSThe parameter that controls the level of diversity in our simulations is m_max, which is the maximal number of different phenotypic clusters allowed to be present at any point in time in an evolving community (see step 4 in the Methods section). Our first result is obtained by allowing this parameter to be very large, so that we can estimate the number of clusters that eventually coexist by simply

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of clusters that eventually coexist by simply running the simulations for a long time and recording the number of clusters at which the diversity equilibrates.We denote by M_,d the equilibrium number of clusters for a given phenotypic dimension d and strength of the Gaussian componentin the competition kernel (<ref>).We found that such equilibrium level of diversity increases exponentially withthe dimension d of phenotype space, and decreases with the strength (Figure 1).Here and below we

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with the strength (Figure 1).Here and below we assume for simplicity that the _i are the same in all phenotypic directions, _i= for i=1,...,d. In theAppendix we indicate scaling relationships that hold for M_,d as functions of the parametersand d. In general, diversity is only maintained if ≲1, which is roughly the scale of the phenotypic range set by the carrying capacity (<ref>). Only if ≲1, the equilibrium level of diversity increases exponentially with increasing dimension of phenotype

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with increasing dimension of phenotype space, Figure 1. Our main results are now obtained based on the observation that by fixing the parameter m_max at a value ≤ M_,d for a given d and , the community will typically evolve to a diversity level m_cluster of approximately m_max. That is, if the diversity is constrained to be below the maximal level of diversity possible for a given set of parameters, then the diversity will typically evolve to the value set by the constraint. Note that this is

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value set by the constraint. Note that this is an “average” statement about many simulations runs, i.e., many different choices of the coefficients b_ij and stochastic realizations of cluster splitting. While some simulation runs will result in a diversity that is lower than m_max (which may reflect an intrinsic state of the system for the given set of coefficients, or a long-living metastable state which has not yet reached its full diversity), most runs will evolve to the level of diversity

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most runs will evolve to the level of diversity that is prescribed by this parameter.This allows us to then assess, for a given level of diversity, the nature of the coevolutionary dynamics that unfolds in communities with that level of diversity. Two paradigmatic examples are shown in Figure 2. We first set the level of diversitym_max=12, which is far below the saturation level M_σ,d for the given system. Starting from very few clusters the diversity quickly evolves to the level set by m_max,

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quickly evolves to the level set by m_max, and the coexisting clusters then exhibit complicated, non-stationary evolutionary dynamics, with all clusters undergoing sustained and irregular fluctuations in phenotype space (Fig. 2a). This type of complicated dynamics is characterized by average evolutionary speeds v>10^-2. In the same system, but now with a value of m_max that lies above the saturation level M_σ,d, the diversity evolves to the saturation level, at which the community consists of

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level, at which the community consists of ca. 30 coexisting phenotypic clusters (Fig. 2b). In this saturated state, the average evolutionary speed is much lower than 10^-2, and the community exhibits much more stationary coevolutionary dynamics (that would eventually converge to a coevolutionary equilibrium). Moreover, the saturated community exhibits a characteristic pattern of over-dispersion in phenotype space due to competitive repulsion caused by the Gaussian component of the competition

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by the Gaussian component of the competition kernel (see also Fig. A1 in the Appendix).To obtain a more systematic characterization of the coevolutionary dynamics as a function of the diversity of the evolving community, we ran, for a given dimension of phenotype space d and strength of competition , 100 simulations with randomly chosen coefficients b_ij for each m_max=1,...,M, where M is some number that is larger than the saturation level of diversity M_,d. For each run, we recorded the

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of diversity M_,d. For each run, we recorded the average per capita evolutionary speed v and the number of phenotypic clusters, i.e., the level of diversity, present at the end of 1000 evolutionary time units (averaged over the last 4 time units). We classified the dynamics into equilibrium dynamics if the average speed v was <10^-2, and non-equilibrium dynamics otherwise. As mentioned earlier, this was based on individual inspection of many simulation that ran longer than 1000 time units,

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simulation that ran longer than 1000 time units, which showed that the threshold 10^-2 is a very good indicator of whether the coevolutionary system eventually equilibrates.Our main results are shown in Figures 3 and 4. The general pattern is that the probability of non-equilibrium dynamics increases as diversity increases from single-cluster communities to communities with a few clusters (Figure 3).For intermediate diversity, the fraction of non-equilibrium dynamics remains high. For

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of non-equilibrium dynamics remains high. For communities with high diversity, the fraction of non-equilibrium dynamics starts to decrease, and almost almost all communities with a diversity close to the saturation level M_,dexhibit equilibrium coevolutionary dynamics.To illustrates these trends, wegive a more detailed account of the average velocities v defined in (<ref>) in the coevolving communities (Fig. 4). It shows that there is an exponential decrease in the average speed as the

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exponential decrease in the average speed as the diversity increases, and that there is a substantial fraction of low-diversity communities that exhibit equilibrium dynamics.The exact shape of these patterns depends on d and(Figures 3 and 4), but whenever diversification is possible, the overall trend is that non-equilibrium dynamics are most likely at intermediate levels of diversity, and that high levels of diversity tend to generate equilibrium coevolutionary dynamics. The patterns shown in

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coevolutionary dynamics. The patterns shown in Figs. 3 and 4 are based on many different simulated communities with different levels of diversity. However, similar patterns can be observed in simulations of single communities as they evolve from low to high diversity, i.e., as they undergo an adaptive radiation. Such a radiation, starting from a single phenotypic cluster, is shown in Fig. 5A.Over time the evolving community becomes more diverse due to adaptive diversification, and as a

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diverse due to adaptive diversification, and as a consequence the nature of the coevolutionary dynamics of the community changes. In the example shown in Figure 5A, the coevolutionary dynamics are fast for low to intermediate levels diversity, and then slow down as the community acquires more and more species, until eventually the community reaches a coevolutionary equilibrium at the diversity saturation level. Again, the slowdown of the evolutionary speed during an adaptive radiation appears

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speed during an adaptive radiation appears to occur exponentially with an increase in diversity. This can also be seen by running a given community defined by a given set of coefficients b_ij for different values of the parameter m_max, determining the level of diversity possible in the evolving community. The evolutionary speed exponentially decreases with the diversity given by m_max (Fig. 5B). We currently do not have a mechanistic explanation for the exponential decay in evolutionary rates

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for the exponential decay in evolutionary rates with increasing diversity. It is informative to watch the process of diversification and subsequent evolutionary slowdown unfold dynamically. To verify that the observed dynamical pattern is not an artifact of the adaptive dynamics approximation, we performed the individual-based and partial differential equation simulations of the same system. The movies in Videos in the Appendix, corresponding to the scenario used for Figures 2B and 5A, confirm

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the scenario used for Figures 2B and 5A, confirm that all three methods produce qualitatively similar evolutionary pictures. The detailed descriptions of the individual-based and partial differential equation methods are given in theAppendix. Another interesting, although perhaps not so surprising observation for single adaptive radiations concerns the rate of accumulation of new species in the evolving ecosystem. Figure 5C shows the number of species as a function of time during the adaptive

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species as a function of time during the adaptive radiation scenario used for Figure 5A, illustrating that the rate of diversification is highest at the beginning of the radiation, and then slows down as the community evolves towards the diversity saturation level. The details of these dynamics depend on system parameters, and in particular on the rate at which new species are introduced into the system, but the qualitative behaviour of diversification rates, which are initially high and then

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rates, which are initially high and then slow down, is common to all adaptive radiations generated by our models.§ DISCUSSION We investigated the expected long-term evolutionary dynamics resulting from competition for resources in models for gradual evolution in high-dimensional phenotype spaces. In reality, most organisms have many different phenotypic properties that impinge on their ecological interactions in generally complicated ways, and here we assumed that multi-dimensional phenotypes

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here we assumed that multi-dimensional phenotypes determine logistic ecological dynamics through the competition kernel and the carrying capacity. We then used a coevolutionary adaptive dynamics algorithm to extend the ecological dynamics to macroevolutionary time scales, and we used a statistical approach to capture general properties of the ensuing evolutionary dynamics.If the negative frequency-dependence generated by the competition kernel is strong enough, competition results in repeated

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is strong enough, competition results in repeated adaptive diversification, and hence in communities of coevolving phenotypic species. By randomly choosing many different competition kernels, we showed that the complexity of the coevolutionary dynamics in such communities is expected to be highest for intermediate levels of phenotypic diversity. In particular, as the evolving communities increase in diversity towards the saturation level, i.e., the maximal number of different species that can

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the maximal number of different species that can coexist, the evolutionary dynamics becomes simpler, and communities at the saturation level are expected to exhibit a coevolutionary equilibrium. We also showed that the diversity saturation level increases exponentially with the dimension of phenotype space.We have used a statistical approach to determine the expected long-term evolutionary dynamics resulting from competition for resources. We have assumed that multi-dimensional phenotypes

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We have assumed that multi-dimensional phenotypes determine logistic ecological dynamics through the competition kernel and the carrying capacity, and we then used a coevolutionary adaptive dynamics algorithm to extend the ecological dynamics to macroevolutionary time scales.If the negative frequency-dependence generated by the competition kernel is strong enough, competition results in repeated adaptive diversification, and hence in communities of coevolving phenotypic species. By randomly

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of coevolving phenotypic species. By randomly choosing many different competition kernels, we showed that the complexity of the coevolutionary dynamics in such communities is expected to be highest for intermediate levels of phenotypic diversity. In particular, as the evolving communities increase in diversity towards the saturation level, i.e., the maximal number of different species that can coexist, the evolutionary dynamics becomes simpler, and communities at the saturation level are

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and communities at the saturation level are expected to exhibit a coevolutionary equilibrium. We also showed that the diversity saturation level increases exponentially with the dimension of phenotype space. Our interpretation of these findings is that in low-dimensional phenotype spaces such as the ones considered here, evolutionary dynamics of single species are expected to converge to an equilibrium <cit.>. However, as diversity increases, the different phenotypic clusters will “push” each

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different phenotypic clusters will “push” each other around evolutionarily due to frequency-dependent competition. This occurs mostly due to the repulsive nature of pairwise interaction induced by the Gaussian term in the competition kernel (<ref>): clusters that move further apart decrease competition felt from each other. For example,a splitting of a cluster stuck in an attractive fixed point of the adaptive dynamics creates two offspring which may become moving again if the repulsion between

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may become moving again if the repulsion between clusters is stronger than the attraction of the fixed point.As long as diversity is not very high, i.e., as long as there is enough available niche or unoccupied phenotype space, this typically results in non-equilibrium coevolutionary dynamics, thus leading to an increase in evolutionary complexity with phenotypic diversity. As the diversity keeps increasing towards saturation levels, which for each phenotypic dimension isdetermined roughly by

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each phenotypic dimension isdetermined roughly by the ratio of the widths of the carrying capacity and the competition kernel (see Video 2), the available carrying capacity niche gets filled, so that the evolving clusters “have nowhere to go” evolutionarily. An analogy with gas-liquid-solid phase transitions may illustrate this in the following way: As in the dynamics of molecules, the adaptive dynamics of phenotypic clusters contains a pairwise-repulsive term, which originates from the

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term, which originates from the Gaussian term in the competition kernel. A few-cluster regime qualitatively corresponds to the gas phase, when the range of the repulsive interaction is significantly less than the typical distance between clusters. As the number and thus density of clusters increases, the repulsive interaction becomes more relevant, constraining the individual motion of clusters and resulting in a liquid-like behaviour, where clusters are predominantly localized and occasionally