Competition, Trait Variance Dynamics, and the Evolution of a Species’ Range

Abstract

Geographic ranges of communities of species evolve in response to environmental, ecological, and evolutionary forces. Understanding the effects of these forces on species’ range dynamics is a major goal of spatial ecology. Previous mathematical models have jointly captured the dynamic changes in species’ population distributions and the selective evolution of fitness-related phenotypic traits in the presence of an environmental gradient. These models inevitably include some unrealistic assumptions, and biologically reasonable ranges of values for their parameters are not easy to specify. As a result, simulations of the seminal models of this type can lead to markedly different conclusions about the behavior of such populations, including the possibility of maladaptation setting stable range boundaries. Here, we harmonize such results by developing and simulating a continuum model of range evolution in a community of species that interact competitively while diffusing over an environmental gradient. Our model extends existing models by incorporating both competition and freely changing intraspecific trait variance. Simulations of this model predict a spatial profile of species’ trait variance that is consistent with experimental measurements available in the literature. Moreover, they reaffirm interspecific competition as an effective factor in limiting species’ ranges, even when trait variance is not artificially constrained. These theoretical results can inform the design of, as yet rare, empirical studies to clarify the evolutionary causes of range stabilization.

Appendices

Appendix A: Model Derivation

To derive the equations of the model given by (1)–(7), we first formulate the intrinsic growth rate of the individuals within each species, which determines the local dynamics of the evolution of the species. For this, at position $x \in \varOmega$ and time $t \in [0,T]$, let $\phi _i(x,t,p)$ denote the relative frequency of a quantitative phenotypic trait with phenotype value $p \in {\mathbb {R}}$ within the ith species. Moreover, let $\alpha _{ij}(p,p')$ denote the competition kernel that captures the per capita effect of phenotype $p'$ in the jth species on the frequency of phenotype p in the ith species. The exact definition of this competition kernel is given in “Competition Kernels” section of “Appendix A” below. Finally, let $g_i(x,t,p)$ denote the intrinsic growth rate of the population of individuals with phenotype p within the ith species.

Intrinsic Growth Rates

For a community of $\mathrm {N}$ competing species, we define the intrinsic growth rate of each species as (Case and Taper 2000, Eq. (2)),

$$\begin{aligned} g_i(x,t,p)&:= \mathrm {R}_i(x) \left( 1 - \frac{1}{\mathrm {K}_i(x)} \sum _{j=1}^{\mathrm {N}} n_j(x,t) \int _{{\mathbb {R}}} \alpha _{ij}(p, p') \phi _j(x,t, p') \mathrm{d} p' \right) \nonumber \\&\quad -\frac{\mathrm {S}}{2} (p - \mathrm {Q}(x))^2, \quad i=1, \dots , \mathrm {N}. \end{aligned}$$

(17)

The first term in (17) is a Lotka–Volterra model of competing species in which the convolution term with $j=i$ expresses the effect of intraspecific phenotypic competition on the frequency of phenotype p, whereas the convolution terms with $j \ne i$ account for the effect of interspecific competition from the phenotypes in the other species. The second term in (17) incorporates the effect of directional and stabilizing selection on individuals with phenotype p. A local population of a species at position x that has a phenotypic trait value different from the environmental optimal value $\mathrm {Q}(x)$ can only reach an equilibrium density that is lower than its carrying capacity. This penalizing effect of the phenotypic selection is made stronger by choosing larger values for $\mathrm {S}$.

Competition Kernels

As proposed by Case and Taper (2000, Eq. (3)), we obtain the competition kernels $\alpha _{ij}$ in (17) using the MacArthur-Levins overlap formula between resource utilization curves of each species (Macarthur and Levins 1967), along with the total resource consumption law given by Roughgarden (1979, Eq. (24.50)). Suppose the environmental resources vary continuously along a resource axis denoted by variable r. Moreover, suppose that individuals with phenotype p within each species possesses a resource utilization curve of the form

$$\begin{aligned} \xi _{i,p}(r) := \varPsi e^{\kappa p} \psi _{i,p}(r), \quad i=1, \dots , \mathrm {N}, \end{aligned}$$

where $\psi _{i,p}(r)$ is a probability density function, which gives the probability density that the individuals obtain a unit of resource from point r. The term $\varPsi e^{\kappa p}$ gives the total amount of resource consumed by an individual with phenotype p. This power law is proposed by Roughgarden (1979, Eq. (24.50)) based on the assumption that energy consumption by an individual is proportional to its weight. This interpretation, however, does not necessarily hold for the general trait-based model presented in this paper, and this specific form of resource consumption form is mainly adopted for the simplicity of derivations and for providing the model with the flexibility of incorporating asymmetric intraspecific competitions with $\kappa \ne 0$.

The resource utilization functions $\xi _{i,p}(r)$ are used to obtain the competition kernels $\alpha _{ij}$ by the following overlap formula (Roughgarden 1979, Eq. (24.5))

$$\begin{aligned} \alpha _{ij}(p,p') := \frac{ \int _{{\mathbb {R}}} \xi _{i,p}(r) \xi _{j,p'}(r) \mathrm{d} r }{\int _{{\mathbb {R}}} \xi _{i,p}^2(r) \mathrm{d} r}. \end{aligned}$$

Calculation of $\alpha _{ij}(p,p')$ based on this formula involves having precise information about resource values. However, it is convenient to assume that the resource axis can be identified by phenotype axis, as proposed by Roughgarden (1979, Eq. (24.51)). For this, let $r_p$ denote the point on the r-axis from which individuals of phenotype p obtain their average amount of resources. We assume that $r_p$ does not depend on which species the individuals belong to, that is, $r_p = \int _{{\mathbb {R}}} r \psi _{i,p}(r) \mathrm{d} r$ for all $i= 1,\dots ,\mathrm {N}$. We further assume that there is a smooth one-to-one map $I:p \mapsto r_p$, which can be used to identify the r-axis with the p-axis, that is, for every $r \in {\mathbb {R}}$ there exist a unique phenotype ${\tilde{p}}\in {\mathbb {R}}$ such that $r=I({\tilde{p}})\equiv {\tilde{p}}$. Therefore, the resource utilization functions and competition kernels can be written only based on trait values, as

$$\begin{aligned} \xi _{i,p}({\tilde{p}})&:= \varPsi e^{\kappa p} \psi _{i,p}({\tilde{p}}),&\quad i=1, \dots , \mathrm {N},&\end{aligned}$$

(18)

$$\begin{aligned} \alpha _{ij}(p,p')&:= \frac{ \int _{{\mathbb {R}}} \xi _{i,p}({\tilde{p}}) \xi _{j,p'}({\tilde{p}}) \mathrm{d} {\tilde{p}} }{\int _{{\mathbb {R}}} \xi _{i,p}^2({\tilde{p}}) \mathrm{d} {\tilde{p}}},&\quad i=1, \dots , \mathrm {N},&\quad j=1, \dots , \mathrm {N}. \end{aligned}$$

(19)

Note that, by the definition of I we have $\int _{{\mathbb {R}}} {\tilde{p}} \psi _{i,p}({\tilde{p}}) \mathrm{d} {\tilde{p}} = p$. We refer to $\xi _{i,p}$ in (18) as phenotype utilization function of individuals with phenotype p within the ith species.

The identification stated above can represent the empirical relationship between functional response traits and environments. In addition to simplifying the mathematical derivations, this identification allows estimation of the parameters of the utilization functions using trait-based approaches to niche quantification (Violle and Jiang 2009; Ackerly and Cornwell 2007). This was further discussed in Sect. 3.

Changes Due to Mutation

Let $\nu (\delta p)$ denote the probability density that by mutation a phenotype p changes to a phenotype $p + \delta p$. We use the equation provided by Kimura (1965), but at the level of phenotypic effects, to approximately model the rate of mutational changes in the frequency of phenotypes as

$$\begin{aligned} \partial _t^{(\mathrm {M})} \phi _i(x,t,p):= -\eta \phi _i(x,t,p) + \eta \int _{{\mathbb {R}}} \nu (p-p') \phi _i(x,t,p') \mathrm{d} p', \quad i=1,\dots ,\mathrm {N},\nonumber \\ \end{aligned}$$

(20)

where $\eta \ge 0$ is the mutation rate per capita per generation. The first term in (20) gives the reduction rate in the frequency of phenotype p within the ith species due to mutation to other phenotypic values. The second term gives the growth rate in the frequency of phenotype p due to mutations to p from other phenotypic values within the ith species.

Model Assumptions

As stated in Sect. 2, the following major assumptions are made on the populations’ dispersal and reproduction, and on the elements of the intrinsic growth rates and competition kernels described above. These assumptions are used in “Derivation of Equations” section of “Appendix A” to derive the equations of the model based on (17).

• (i) Each species disperses in the habitat by diffusion.

• (ii) Nonlinear environmental selection for the optimal phenotype $\mathrm {Q}(x)$ is stabilizing for all $x\in \varOmega$.

• (iii) Frequency of trait values follows a normal distribution for all $x \in \varOmega$ and $t \in [0,T]$, that is,

  $$\begin{aligned} \phi _i(x,t,p) := \frac{1}{\sqrt{2 \pi v_i(x,t)}} \exp \left( -\frac{(p - q_i(x,t))^2}{2 v_i(x,t)} \right) , \quad i =1,\dots , \mathrm {N}.\quad \end{aligned}$$

  (21)

• (iv) Within each species, the reproduction rate of individuals with phenotype p depends on the population density of individuals with the same phenotype p.

• (v) Phenotype utilization distribution $\psi _{i,p}$ in (18) is normal, that is,

  $$\begin{aligned} \psi _{i,p}({\tilde{p}}) = \frac{1}{\sqrt{2 \pi \mathrm {V}_i}} \exp \left( -\frac{({\tilde{p}} - p)^2}{2 \mathrm {V}_i} \right) , \quad i =1,\dots , \mathrm {N}. \end{aligned}$$

  (22)

  Therefore, competition kernels given by (19) can be calculated as

  $$\begin{aligned} \alpha _{ij}(p,p')= & {} \varLambda _{ij} \exp (\kappa ^2 {\bar{\mathrm {V}}}_{ij}) \exp \left( -\frac{(p - p' + 2 \kappa {\bar{\mathrm {V}}}_{ij})^2}{4 {\bar{\mathrm {V}}}_{ij}} \right) ,\nonumber \\&\qquad \qquad \qquad \qquad i=1, \dots , \mathrm {N}, \quad j=1, \dots , \mathrm {N}, \end{aligned}$$

  (23)

  where $\varLambda _{ij}:=\sqrt{\smash [b] { \mathrm {V}_i / {\bar{\mathrm {V}}}_{ij}}}$ with ${\bar{\mathrm {V}}}_{ij} := \frac{1}{2}(\mathrm {V}_i + \mathrm {V}_j)$, as in (7).

• (vi) The mutation kernel $\nu$ in (20) is the probability density function of a probability distribution with constant zero mean and constant variance $\mathrm {V}_{\textsc {m}}$. That is, in particular, $\nu$ is independent of population density, trait mean, or baseline trait variance.

Remark 6

(Symmetric competition kernel) The MacArthur-Levins overlap formula (19) gives asymmetric competition kernels (23), wherein $\alpha _{ij}(p,p') \ne \alpha _{ji}(p,p')$ when $\mathrm {V}_i \ne \mathrm {V}_j$ or $\kappa \ne 0$. A symmetric alternative to (19) is proposed in the literature (Pianka 1973), which can be written as

$$\begin{aligned} \alpha _{ij}(p,p') := \frac{ \int _{{\mathbb {R}}} \xi _{i,p}({\tilde{p}}) \xi _{j,p'}({\tilde{p}}) \mathrm{d} {\tilde{p}} }{\left( \int _{{\mathbb {R}}} \xi _{i,p}^2({\tilde{p}}) \mathrm{d} {\tilde{p}} \int _{{\mathbb {R}}} \xi _{i,p'}^2({\tilde{p}}) \mathrm{d} {\tilde{p}} \right) ^{\frac{1}{2}} }, \quad i=1, \dots , \mathrm {N}, \quad j=1, \dots , \mathrm {N}. \end{aligned}$$

With the normal density function $\psi _{i,p}$ given in (22), this symmetric overlap formula yields

$$\begin{aligned} \alpha _{ij}(p,p') = \varLambda _{ij} \exp \left( -\frac{(p - p')^2}{4 {\bar{\mathrm {V}}}_{ij}} \right) , \quad i=1, \dots , \mathrm {N}, \quad j=1, \dots , \mathrm {N}, \end{aligned}$$

(24)

where $\varLambda _{ij}:=\sqrt{\mathring{\mathrm {V}}_{ij} /{\bar{\mathrm {V}}}_{ij}}$ with $\mathring{\mathrm {V}}_{ij} := \sqrt{\mathrm {V}_i \mathrm {V}_j}$ and ${\bar{\mathrm {V}}}_{ij} := \frac{1}{2}(\mathrm {V}_i + \mathrm {V}_j)$. In “Derivation of Equations” section of “Appendix A,” however, the asymmetric kernels (23) are used to derive the equations of the model given in (7). Note that, (23) can be easily transformed to (24) by setting $\kappa =0$ and replacing $\mathrm {V}_i$ with $\mathring{\mathrm {V}}_{ij}$. $\square$

Derivation of Equations

The derivation of Eqs. (1)–(7) begins with the following equation

$$\begin{aligned}&n_i(x,t+\tau ) \phi _i(x,t+\tau ,p) - n_i(x,t)\phi _i(x,t,p) \nonumber \\&\quad = \tau \big [ {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t)\phi _i(x,t,p))) + g_i(x,t,p) n_i(x,t)\phi _i(x,t,p) \nonumber \\&\qquad + n_i(x,t) \partial _t^{(\mathrm {M})} \phi _i(x,t,p) \big ], \quad i=1, \dots , \mathrm {N}, \end{aligned}$$

(25)

wherein, within each species the variation in the population density of individuals with phenotype p over a small time interval $\tau \rightarrow 0$ is assumed to result from the contributions of three factors, namely, the diffusive migration of individuals to and from neighboring locations, the intrinsic growth of the population, and the mutational changes in the relative frequency of p.

Integrating both sides of (25) with respect to p over ${\mathbb {R}}$, we obtain

$$\begin{aligned} n_i(x,t+\tau ) - n_i(x,t)= & {} \tau \big [ {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t))) \nonumber \\&+ G_i(x,u(x,t)) n_i(x,t) \big ], \end{aligned}$$

(26)

where

$$\begin{aligned} G_i(x,u(x,t)) := \int _{{\mathbb {R}}} g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p \end{aligned}$$

(27)

denotes the mean value of the intrinsic growth rate of the population of individuals with phenotype p within the ith species. Note that in writing (27) we have used (20) to obtain $\int _{{\mathbb {R}}} \partial _t^{(\mathrm {M})} \phi _i(x,t,p) \mathrm{d} p =0$. Moreover, we have implicitly presumed that the mean value of $g_i(x,t,p)$ can be written in terms of x and the variables of the model, u. This is indeed true by the calculations that follow below. In addition, note that (1) is obtained immediately by dividing both sides of (26) by $\tau$ and taking the limit as $\tau \rightarrow 0$, provided $G_i(x,u)$ is shown to be given by (4).

Next, to derive (2), we multiply both sides of (25) by p and integrate the result with respect to p over ${\mathbb {R}}$. Note that the zero-mean assumption (vi) on the mutation distribution, along with (20), gives $\int _{{\mathbb {R}}} p \partial _t^{(\mathrm {M})} \phi _i(x,t,p) \mathrm{d} p =0$. Therefore, we obtain

$$\begin{aligned}&n_i(x,t+\tau ) q_i(x,t+\tau ) - n_i(x,t)q_i(x,t) \nonumber \\&\quad = \tau \left[ {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t)q_i(x,t))) + n_i(x,t) \int _{{\mathbb {R}}} p g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p \right] , \end{aligned}$$

(28)

which further implies, after dividing by $\tau$ and taking the limit as $\tau \rightarrow 0$, that

$$\begin{aligned} \partial _t(n_i(x,t) q_i(x,t))= & {} {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t)q_i(x,t))) \nonumber \\&+\, n_i(x,t) \int _{{\mathbb {R}}} p g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p. \end{aligned}$$

(29)

Now, using the chain rule on the left hand side of the above equation and substituting (1) into the result, we obtain

$$\begin{aligned} \partial _tq_i(x,t)&= \frac{1}{n_i(x,t)} \Big [ {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t)q_i(x,t))) \\&\quad - q_i(x,t) \big ( {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _xn_i(x,t)) + G_i(x,u(x,t)) n_i(x,t) \big ) \Big ]\\&\quad + \int _{{\mathbb {R}}} p g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p. \end{aligned}$$

For the first term within the brackets we can write

$$\begin{aligned}&{{\,\mathrm{div}\,}}( \mathrm {D}_i(x) \partial _x(n_i(x,t)q_i(x,t)) ) \\&\quad = {{\,\mathrm{div}\,}}( q_i(x,t) \mathrm {D}_i(x) \partial _xn_i(x,t) ) + {{\,\mathrm{div}\,}}( n_i(x,t) \mathrm {D}_i(x) \partial _xq_i(x,t) ) \\&\quad =q_i(x,t) {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _xn_i(x,t) ) + 2 \left( \partial _xn_i(x,t) , \mathrm {D}_i(x) \partial _xq_i(x,t) \right) _{{\mathbb {R}}^{\mathrm {m}}} \\&\qquad + n_i(x,t) {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _xq_i(x,t) ). \end{aligned}$$

Therefore, it follows that

$$\begin{aligned} \partial _tq_i(x,t)= & {} {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _xq_i(x,t)) + 2 \left( \frac{\partial _xn_i(x,t)}{n_i(x,t)} , \mathrm {D}_i(x) \partial _xq_i(x,t) \right) _{{\mathbb {R}}^{\mathrm {m}}} \!\!\\&+ H_i(x,u(x,t)), \end{aligned}$$

where

$$\begin{aligned} H_i(x,u(x,t)):= \int _{{\mathbb {R}}} p g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p - G_i(x,u(x,t)) q_i(x,t). \end{aligned}$$

(30)

This gives (2), provided we show $H_i(x,u)$ can be given by (5).

Finally, to derive (3), we multiply both sides of (25) by $(p-q_i(x,t+\tau ))^2$ and integrate the result with respect to p over ${\mathbb {R}}$. For the mutation term in (25), it follows from (20) and assumption (vi) that $\int _{{\mathbb {R}}} (p-q_i(x,t+\tau ))^2 \partial _t^{(\mathrm {M})} \phi _i(x,t,p) \mathrm{d} p = \mathrm {U}$, where $\mathrm {U}:= \eta \mathrm {V}_{\textsc {m}}$. Therefore, we obtain

$$\begin{aligned}&n_i(x,t+\tau ) v_i(x,t+\tau ) \\&\quad = \int _{{\mathbb {R}}} (p-q_i(x,t+\tau ))^2 \big [ \tau {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t)\phi _i(x,t,p))) \\&\qquad + (1 + \tau g_i(x,t,p)) n_i(x,t)\phi _i(x,t,p) \big ] \mathrm{d} p + \tau {n_i(x,t) \mathrm {U}} \\&\quad = \tau \int _{{\mathbb {R}}} p^2 \big [ {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t)\phi _i(x,t,p))) + g_i(x,t,p) n_i(x,t) \phi _i(x,t,p) \big ] \mathrm{d} p \\&\qquad + n_i(x,t) \int _{{\mathbb {R}}} p^2 \phi _i(x,t,p) \mathrm{d} p - n_i(x,t+\tau ) q_i^2(x,t+\tau ) + \tau {n_i(x,t) \mathrm {U}} \\&\quad = \tau \left[ {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t) v_i(x,t))) + {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t) q_i^2(x,t))) \right] \\&\qquad + \tau n_i(x,t) \int _{{\mathbb {R}}} p^2 g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p + n_i(x,t) v_i(x,t) \\&\qquad - \left( n_i(x,t+\tau ) q_i^2(x,t+\tau ) - n_i(x,t) q_i^2(x,t) \right) + \tau {n_i(x,t) \mathrm {U}}. \end{aligned}$$

Dividing both sides of the above equation by $\tau$ and taking the limit as $\tau \rightarrow 0$, we obtain

$$\begin{aligned}&\partial _t(n_i(x,t)v_i(x,t)) \nonumber \\&\quad = {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t) v_i(x,t))) \nonumber \\&\qquad + {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t) q_i^2(x,t))) \nonumber \\&\qquad + n_i(x,t) \int _{{\mathbb {R}}} p^2 g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p - \partial _t(n_i(x,t) q_i^2(x,t)) +{n_i(x,t) \mathrm {U}}. \end{aligned}$$

(31)

Note that,

$$\begin{aligned} {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t) q_i^2(x,t)))&= q_i^2(x,t){{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _xn_i(x,t) )\\&\quad + 2 n_i(x,t) q_i(x,t) {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _xq_i(x,t)) \\&\quad + 2 n_i(x,t) \left( \partial _xq_i(x,t) , \mathrm {D}_i(x) \partial _xq_i(x,t) \right) _{{\mathbb {R}}^{\mathrm {m}}} \\&\quad + 4 q_i(x,t) \left( \partial _xn_i(x,t) , \mathrm {D}_i(x) \partial _xq_i(x,t) \right) _{{\mathbb {R}}^{\mathrm {m}}}. \end{aligned}$$

Moreover, $\partial _t(n_i(x,t) q_i^2(x,t))$ can be calculated using the chain rule and Eqs. (1) and (2), wherein $G_i(x,u)$ and $H_i(x,u)$ are given by (27) and (30), respectively. Therefore, (31) gives

$$\begin{aligned} \partial _t(n_i(x,t)v_i(x,t))&= {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t) v_i(x,t)))\nonumber \\&\quad + n_i(x,t) \bigg [2 \left( \partial _xq_i(x,t) , \mathrm {D}_i(x) \partial _xq_i(x,t) \right) _{{\mathbb {R}}^{\mathrm {m}}} \nonumber \\&\quad + \int _{{\mathbb {R}}} p^2 g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p\nonumber \\&\quad -2 q_i(x,t) \int _{{\mathbb {R}}} p g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p \nonumber \\&\quad + G_i(x,u(x,t)) q_i^2(x,t) { +\mathrm {U}}\bigg ]. \end{aligned}$$

(32)

Now, note that

$$\begin{aligned} {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _x(n_i(x,t) v_i(x,t)))&= v_i(x,t) {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _xn_i(x,t) ) \\&\quad + 2 \left( \partial _xn_i(x,t) , \mathrm {D}_i(x) \partial _xv_i(x,t) \right) _{{\mathbb {R}}^{\mathrm {m}}} \\&\quad + n_i(x,t) {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _xv_i(x,t) ). \end{aligned}$$

Therefore, using the chain rule on the left hand side of (32) and substituting (1) into the result, we obtain

$$\begin{aligned} \partial _tv_i(x,t)&= {{\,\mathrm{div}\,}}(\mathrm {D}_i(x) \partial _xv_i(x,t) ) + 2 \left( \frac{\partial _xn_i(x,t)}{n_i(x,t)} , \mathrm {D}_i(x) \partial _xv_i(x,t) \right) _{{\mathbb {R}}^{\mathrm {m}}}\\&\quad + 2 \left( \partial _xq_i(x,t) , \mathrm {D}_i(x) \partial _xq_i(x,t) \right) _{{\mathbb {R}}^{\mathrm {m}}} + W_i(x,u(x,t)), \end{aligned}$$

where

$$\begin{aligned} W_i(x,u(x,t))&:= \int _{{\mathbb {R}}} p^2 g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p \nonumber \\&\quad -2 q_i(x,t) \int _{{\mathbb {R}}} p g_i(x,t,p) \phi _i(x,t,p) \mathrm{d} p \nonumber \\&\quad - G_i(x,u(x,t)) \left( v_i(x,t) - q_i^2(x,t) \right) { +\mathrm {U}}. \end{aligned}$$

(33)

This gives (3) provided $W_i(x,u)$ is shown to be given by (6).

Now, to complete the derivation of (1)–(3), it remains to show that $G_i(x,u)$, $H_i(x,u)$, and $W_i(x,u)$ can be given by (4), (5), and (6), respectively. For simplicity of exposition, the dependence of functions $n_i$, $q_i$, $v_i$, $g_i$, and $\phi _i$ on variables x and t, as well as the dependence of $\mathrm {R}_i$, $\mathrm {K}_i$, and $\mathrm {Q}$ on x, are not explicitly shown in the rest of this section.

We begin with calculating $g_j(p)$. Using (21) and (23), the integral in (17) can be written as

$$\begin{aligned}&\int _{{\mathbb {R}}} \alpha _{ij}(p,p') \phi _j(p') \mathrm{d} p'\nonumber \\&\quad = \frac{\varLambda _{ij}\exp (\kappa ^2 {\bar{\mathrm {V}}}_{ij})}{\sqrt{2\pi v_j}} \int _{{\mathbb {R}}} \exp \left( -\frac{(p -p' + 2 \kappa {\bar{\mathrm {V}}}_{ij})^2}{4{\bar{\mathrm {V}}}_{ij}} \right) \exp \left( -\frac{(p' - q_j)^2}{2 v_j} \right) \mathrm{d} p' \nonumber \\&\quad = \frac{\varLambda _{ij}\exp (\kappa ^2 {\bar{\mathrm {V}}}_{ij})}{\sqrt{2\pi v_j}} {\hat{M}}_j(u) \int _{{\mathbb {R}}} {\hat{A}}_j(p',u) \mathrm{d} p', \end{aligned}$$

(34)

where

$$\begin{aligned} {\hat{M}}_j(u)&:=\exp \left( -\frac{ -\dfrac{[2 v_j (p + 2 \kappa {\bar{\mathrm {V}}}_{ij}) + 4 {\bar{\mathrm {V}}}_{ij} q_j ]^2 }{2 v_j + 4 {\bar{\mathrm {V}}}_{ij}} + 2 v_j (p + 2 \kappa {\bar{\mathrm {V}}}_{ij})^2 + 4 {\bar{\mathrm {V}}}_{ij} q_j^2 }{(4{\bar{\mathrm {V}}}_{ij}) (2 v_j)} \right) \nonumber \\&= \exp \left( -\frac{[q_j - (p + 2 \kappa {\bar{\mathrm {V}}}_{ij})]^2}{2 v_j + 4 {\bar{\mathrm {V}}}_{ij}} \right) , \end{aligned}$$

(35)

and

$$\begin{aligned} {\hat{A}}_j(p',u) :=\exp \left( -\frac{(p' - {\hat{\mu }}_j(u))^2}{2 {\hat{\sigma }}_j^2(u)} \right) , \end{aligned}$$

(36)

with

$$\begin{aligned} {\hat{\mu }}_j(u):= \frac{2 v_j (p + 2 \kappa {\bar{\mathrm {V}}}_{ij}) + 4 {\bar{\mathrm {V}}}_{ij} q_j}{2 v_j + 4 {\bar{\mathrm {V}}}_{ij}}, \quad {\hat{\sigma }}_j^2(u):= \frac{1}{2} \frac{(4 {\bar{\mathrm {V}}}_{ij})(2v_j)}{2 v_j + 4 {\bar{\mathrm {V}}}_{ij}}. \end{aligned}$$

(37)

Note that $\int _{{\mathbb {R}}} {\hat{A}}_j(p',u) \mathrm{d} p' = \sqrt{2 \pi }{\hat{\sigma }}_j(u)$. Therefore, substituting the results into (17), we obtain

$$\begin{aligned} g_i(p)&= \mathrm {R}_i - \frac{\mathrm {R}_i}{\mathrm {K}_i} \sum _{j=1}^{\mathrm {N}} \frac{\sqrt{2 {\bar{\mathrm {V}}}_{ij}} \varLambda _{ij} \exp (\kappa ^2 {\bar{\mathrm {V}}}_{ij})}{\sqrt{\smash [b]{v_j + 2 {\bar{\mathrm {V}}}_{ij}}}} \exp \left( -\frac{[q_j - (p + 2 \kappa {\bar{\mathrm {V}}}_{ij})]^2}{2 v_j + 4 {\bar{\mathrm {V}}}_{ij}} \right) n_j \nonumber \\&\quad -\frac{\mathrm {S}}{2}(p-\mathrm {Q})^2. \end{aligned}$$

(38)

Now, we substitute (38) into (27) to calculate $G_i(x,u)$. Note that,

$$\begin{aligned} \int _{{\mathbb {R}}} \mathrm {R}_i \phi _i(p) \mathrm{d} p&= \mathrm {R}_i, \end{aligned}$$

(39)

$$\begin{aligned} \int _{{\mathbb {R}}} \frac{\mathrm {S}}{2} (p-\mathrm {Q})^2 \phi _i(p) \mathrm{d} p&= \frac{\mathrm {S}}{2 } \int _{{\mathbb {R}}} (p^2 - 2 p \mathrm {Q}+ \mathrm {Q}^2) \phi _i(p) \mathrm{d} p \nonumber \\&= \frac{\mathrm {S}}{2} \left[ (v_i + q_i^2) - 2 q_i \mathrm {Q}+ \mathrm {Q}^2) \right] \nonumber \\&= \frac{\mathrm {S}}{2} \left[ (q_i - \mathrm {Q})^2 + v_i \right] . \end{aligned}$$

(40)

Moreover, the integral associated with the term inside the summation in (38) can be calculated using similar calculation as given for (34). Specifically, as compared with (34)–(37),

$$\begin{aligned}&\frac{1}{\sqrt{2 \pi v_i}} \int _{{\mathbb {R}}} \exp \left( -\frac{[q_j - (p + 2 \kappa {\bar{\mathrm {V}}}_{ij})]^2}{2 v_j + 4 {\bar{\mathrm {V}}}_{ij}} \right) \exp \left( -\frac{(p - q_i)^2}{2 v_i} \right) \mathrm{d} p \nonumber \\&\quad = \frac{M_{ij}(u)}{\sqrt{2 \pi v_i}} \int _{{\mathbb {R}}} A_{ij}(p,u) \mathrm{d} p, \end{aligned}$$

(41)

where $M_{ij}(u)$ is given in (7) and

$$\begin{aligned} A_{ij}(p,u) :=\exp \left( -\frac{(p - \mu _{ij}(u))^2}{2 \sigma _{ij}^2(u)} \right) , \end{aligned}$$

(42)

with

$$\begin{aligned} \mu _{ij}(u):= \frac{2 v_i (q_j - 2 \kappa {\bar{\mathrm {V}}}_{ij}) + (2 v_j + 4 {\bar{\mathrm {V}}}_{ij}) q_i}{2 v_i + (2 v_j + 4 {\bar{\mathrm {V}}}_{ij})}, \quad \sigma _{ij}^2(u):= \frac{1}{2} \frac{(2 v_j + 4 {\bar{\mathrm {V}}}_{ij})(2v_i)}{2 v_i + (2 v_j + 4 {\bar{\mathrm {V}}}_{ij})}. \end{aligned}$$

(43)

Now, note that $\int _{{\mathbb {R}}} A_{ij}(p,u) \mathrm{d} p = \sqrt{2 \pi } \sigma _{ij}(u)$. Therefore, substituting (38) into (27), using (39)–(43), and letting $C_{ij}(u)$ be defined as in (7), the mean growth rate $G_i(x,u)$ is obtained as given by (4).

Next, we substitute (38) into (30) to calculate $H_i(x,u)$. We can write

$$\begin{aligned} \int _{{\mathbb {R}}} p \mathrm {R}_i \phi _i(p) \mathrm{d} p&= \mathrm {R}_i q_i, \end{aligned}$$

(44)

$$\begin{aligned} -\int _{{\mathbb {R}}} p \frac{\mathrm {S}}{2}(p-\mathrm {Q})^2 \phi _i(p) \mathrm{d} p&= -\frac{\mathrm {S}}{2} \int _{{\mathbb {R}}} (p^3 - 2 p^2 \mathrm {Q}+ p \mathrm {Q}^2 ) \phi _i(p) \mathrm{d} p \nonumber \\&= - \frac{\mathrm {S}}{2} \left[ (3 v_i q_i + q_i^3) - 2 (v_i + q_i^2) \mathrm {Q}+ q_i \mathrm {Q}^2 ) \right] . \end{aligned}$$

(45)

Note that (45) is equal to $E_i(x,u)$ as defined in (7). Moreover, as compared with (34)–(37),

$$\begin{aligned}&\frac{1}{\sqrt{2 \pi v_i}} \int _{{\mathbb {R}}} p \exp \left( -\frac{[q_j - (p + 2 \kappa {\bar{\mathrm {V}}}_{ij})]^2}{2 v_j + 4 {\bar{\mathrm {V}}}_{ij}} \right) \exp \left( -\frac{(p - q_i)^2}{2 v_i} \right) \mathrm{d} p\nonumber \\&\quad = \frac{M_{ij}(u)}{\sqrt{2 \pi v_i}} \int _{{\mathbb {R}}} p A_{ij}(p,u) \mathrm{d} p, \end{aligned}$$

(46)

where $M_{ij}(u)$ and $A_{ij}(p,u)$ are given by (7) and (42), respectively. Therefore, we have $\int _{{\mathbb {R}}} p A_{ij}(p,u) \mathrm{d} p = \sqrt{2 \pi } \sigma _{ij}(u) \mu _{ij}(u)$, where $\mu _{ij}(u)$ and $\sigma _{ij}(u)$ are given by (43). Now, letting $L_{ij}(u)$ be defined as in (7), Eq. (30) along with (38) and (43)–(46) gives $H_i(x,u)$ as in (5).

Finally, we use (33) with (38) and (30) to calculate $W_i(x,u)$. Note that,

$$\begin{aligned} \int _{{\mathbb {R}}} p^2 \mathrm {R}_i \phi _i(p) \mathrm{d} p&= (v_i + q_i^2) \mathrm {R}_i , \end{aligned}$$

(47)

$$\begin{aligned} -\int _{{\mathbb {R}}} p^2 \frac{ \mathrm {S}}{2} (p-\mathrm {Q})^2 \phi _i(p) \mathrm{d} p&= - \frac{\mathrm {S}}{2} \int _{{\mathbb {R}}} (p^4 - 2 p^3 \mathrm {Q}+ p^2 \mathrm {Q}^2 ) \phi _i(p) \mathrm{d} p \nonumber \\&= -\frac{\mathrm {S}}{2} \left[ (3 v_i^2 + 6 v_i q_i^2 + q_i^4) \right. \nonumber \\&\quad \left. - 2 (3 v_i q_i + q_i^3) \mathrm {Q}+ (v_i + q_i^2) \mathrm {Q}^2 ) \right] \nonumber \\&=: {\hat{Y}}_i(x,u), \end{aligned}$$

(48)

and, as compared with (34)–(37),

$$\begin{aligned}&\frac{1}{\sqrt{2 \pi v_i}} \int _{{\mathbb {R}}} p^2 \exp \left( -\frac{[q_j - (p + 2 \kappa {\bar{\mathrm {V}}}_{ij})]^2}{2 v_j + 4 {\bar{\mathrm {V}}}_{ij}} \right) \exp \left( -\frac{(p - q_i)^2}{2 v_i} \right) \mathrm{d} p\nonumber \\&\quad = \frac{M_{ij}(u)}{\sqrt{2 \pi v_i}} \int _{{\mathbb {R}}} p^2 A_{ij}(p,u) \mathrm{d} p, \end{aligned}$$

(49)

where $M_{ij}(u)$ and $A_{ij}(p,u)$ are given by (7) and (42), respectively. It follows that $\int _{{\mathbb {R}}} p^2 A_{ij}(p,u) \mathrm{d} p = \sqrt{2 \pi } \sigma _{ij}(u) [\sigma _{ij}^2(u) + \mu _{ij}^2(u)]$, where $\mu _{ij}(u)$ and $\sigma _{ij}(u)$ are given by (43). Therefore, the first integral in (33) can be written as

$$\begin{aligned} \int _{{\mathbb {R}}} p^2 g_i(p) \phi _i(p) \mathrm{d} p = (v_i + q_i^2) \mathrm {R}_i -\frac{\mathrm {R}_i}{\mathrm {K}_i} \sum _{j=1}^{\mathrm {N}} {\hat{P}}_{ij}(u) M_{ij}(u) C_{ij}(u) + {\hat{Y}}_i(x,u)\nonumber \\ \end{aligned}$$

(50)

where ${\hat{Y}}_i(x,u)$ is given by (48) and

$$\begin{aligned} {\hat{P}}_{ij}(u) := \frac{v_i(v_j + 2 {\bar{\mathrm {V}}}_{ij})}{v_i + v_j + 2 {\bar{\mathrm {V}}}_{ij}} + L_{ij}^2(u). \end{aligned}$$

(51)

Moreover, the second integral in (33) can be calculated immediately using (30) and (5). The result along with (50) gives $W_i(x,u)$ as in (6), wherein $P_{ij}(u) = {\hat{P}}_{ij}(u) - 2 q_i L_{ij}(u)$ and $Y_i(x,u) = {\hat{Y}}_i(x,u) - 2 q_i E_i(x,u) { +\mathrm {U}}$. Note that, using (48) and (51), we obtain $P_{ij}(u)$ and $Y_i(x,u)$ as given in (7). This completes the derivation of the model.

Finally, the homogeneous Neumann boundary conditions (8) are obtained by assuming no phenotypic flux through the boundaries, that is,

$$\begin{aligned}&\mathrm {D}_i(x) \partial _x( n_i(x,t) \phi _i(x,t,p) ) = 0, \quad i=1,\dots , \mathrm {N}, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \text {for all } (x,t)\in \{a,b\}\times [0,T] \text { and all } p \in {\mathbb {R}}.\qquad \end{aligned}$$

(52)

Integrating this condition with respect to p over ${\mathbb {R}}$ and noting that in general $\mathrm {D}_i$ is nonzero on the boundary, we obtain the boundary condition $\partial _xn_i = 0$ as in (8). Moreover, it follows from multiplying (52) by p, integrating the result over ${\mathbb {R}}$, and using the condition $\partial _xn_i = 0$, that $\partial _x(n_i q_i) = n_i \partial _xq_i = 0$ on the boundary. This gives the boundary condition $\partial _xq_i = 0$ given in (8), since $n_i$ is not required to be zero on the boundary under a no-flux condition. The boundary condition $\partial _xv_i = 0$ given in (8) is obtained similarly by multiplying (52) by $(p - q_i(x,t))^2$ and integrating the result over ${\mathbb {R}}$.

Appendix B: Numerical Methods and Discretization Parameters

The numerical solutions presented in Sects. 4 and 5 have been computed using an Alternating Direction Implicit (ADI) scheme with two stabilizing correction stages, as presented by Hundsdorfer (2002, Eq. (20)). The parameter $\theta$ in the formulation of this scheme is set to $\theta = 1/2$. The function F in the formulation has two components when we consider a one-dimensional geographic space. One of these components is associated with the terms involving spatial derivatives, and the other component is associated with the reaction terms. For the two-dimensional space considered in Sect. 5.4, the function F has three components, first component associated with derivatives with respect to $x_1$, second component associated with derivatives with respect to $x_2$, and third component associated with the reaction terms. We treated all components in both spatial dimensions implicitly. For further details of this numerical method, see the results developed by Hundsdorfer (2002) and in ’t Hout and Welfert (2007).

In each iteration of the scheme, instead of solving the nonlinear algebraic equations of the scheme using Newton’s method, we have solved the linearized version of these equations. This is in fact equivalent to performing only one Newton iteration. The required changes in the formulations to incorporate this linearization step are also provided by Hundsdorfer (2002). The computational time that is saved by solving linearized equations can then be used to allow for smaller time steps, which can in turn compensate for the loss of accuracy due to the linearization. Linearizing the scheme has the advantage of providing a better control over the total computation time of the simulations, as the computation time will then become almost linearly proportional to the time steps.

Finally, we have used fourth-order centered difference approximations for both first and second derivatives in each spatial direction. In one-dimensional space, we have considered a uniform discretization mesh of size $\varDelta x = 0.1 \, \mathtt {X}$, as well as uniform time steps of length $\varDelta t = 0.002 \, \mathtt {T}$. For the two-dimensional problem of Sect. 5.4, we have used a rectangular mesh of size $\varDelta x_1 \times \varDelta x_2 = 0.5 \times 0.5 \, \mathtt {X}^2$, and a time step of $\varDelta t = 0.01 \, \mathtt {T}$.