Space is subdivided into a rectangular array of “patches”. There are \(2\) environmental characteristics corresponding to the height (\(u\)) at which a lizard occurs and the diameter (\(v\)) of the object upon which they are sitting. Each patch is characterized by a frequency distribution \(F(u,v)\). We can consider 2 possibilities.
Old version.The distribution of \(u\) is normal with mean \(\mu\) and variance \(\sigma^2\), and \(v\) has an exponential distribution with mean \(\theta^{-1} = c_0 - c_1 u\), where \(c_0\) and \(c_1\) are positive constants. In particular, the natural constraint \(\theta^{-1} > 0\) implies \(1 \le c_0/c_1\). The joint density \(F\) is \[\label{gaus} F(u,v) \; = \; k \,\exp \big(-\sigma (u - \mu)^2 \big) \exp\big(\!- \theta v \big) \theta,\] where \(k\) is normalizing constant. We can truncate this distribution so it’s defined for \(0\leq u,v \leq 1\). We can also set the mean of \(u\) at \(\mu=1/2\).
New version.Height \(u\) has a symmetric beta distribution mean 1/2 and with parameter \(\alpha>1\) and the diameter has a triangular distribution on the interval \([0,1-u]\) so that the joint density is \[\label{beta} F(u,v) \; = \; \frac{\Gamma(2\alpha)}{\Gamma(\alpha)^2} \big(u(1-u)\big)^{\alpha-1}\ \frac{2}{1-u}(1-\frac{v}{1-u}),\] where \(\Gamma\) is gamma function. The mean of \(u\) is 1/2 and the variance is \(1/(4(1+2\alpha))\). For any given \(u\), the mean of \(v\) is \((1-u)/3\). This distribution is defined for \(0 \leq u \leq 1\) and \(0 \leq v \leq 1-u\). The triangular distribution seems to be a good approximation of the exponential distribution (good for our purposes).
Individuals are sexual and diploid. Each individual has a number of additive quantitative characters:
2 “morphological” characters \(x_0\) (toepad size) and \(x_1\) (limb length) that control viability;
2 “habitat preference” characters: the most preferred height \(y_0\) and the most preferred diameter \(y_1\)
three “mating compatibility” characters \(m, f\), and \(c\).
The male display trait \(m\) is expressed in males only, whereas female mating preference \(f\) and tolerance \(c\) are expressed in females only. Other traits are expressed in both sexes. All traits are scaled to be between 0 and 1 and are controlled by different unlinked diallelic loci with equal effects. Mutations occur at equal rates across all loci; the probabilities of forward and backward mutations are equal.
Habitat preference.The relative preference of an individual with habitat preference traits \(y_0,y_1\) for environmental characteristic \((u,v)\) is \[\Xi(y_0,y_1\mid u,v) = \exp\big(-\frac{(u-y_0)^2}{2h_0^2} -\frac{(v-y_1)^2}{2h_1^2}\big)\] We can also consider a quadratic approximation: \[\label{quad} \Xi(y_0,y_1\mid u,v) = 1-h_0(u-y_0)^2-h_1(v-y_1)^2.\]
Viability.Let viability of genotype \((x_0,x_1)\) in niche \((u,v)\) be Gaussian, \[W(x_0,x_1 \mid u,v) = \exp\big(-\frac{(u-x_0)^2}{2s_0^2} -\frac{(v-x_1)^2}{2s_1^2}\big),\] where the \(\sigma_{s_i}>0\) are parameters measuring the strength of selection. Note that the optimum values of traits \(x_0\) and \(x_1\) under environmental conditions \((u,v)\) are \(x_0 = u\) and \(x_1 = v\).
Carrying capacity.Consider a specific genotype that has an absolute preference for environment \((u,v)\) (so that \(\Xi\) is a \(\delta\)-function). Then a natural measure of the carrying capacity of a population of such individuals is \[K_{\max} F(u,v) W(x_0,x_1 \mid u,v),\] where \(K_{\max}\) is a scaling parameter, and \(K_{\max} F(u,v)\) can be thought of as the maximum number of individuals with an absolute preference for environment \((u,v)\) is they also have perfect adaptation for this niche (i.e. if \(W(x_0,x_1 \mid u,v)=1\)). Naturally, the more common environments can maintain higher densities of individuals that prefer them. Now, \[K_e(I)= K_{\max} \iint F(u,v) W(x_0,x_1 \mid u,v) \Xi(y_0,y_1\mid u,v) du dv\] is the average carrying capacity of genotype \((x_0,x_1, y_0, y_1)\).
Following Roughgarden and others, we define the competition coefficient for individual \(I\) and \(J\) as
\[\begin{aligned} C(I,J) &= C_x(I,J) C_y(I,J)\\ C_x(I,J) &= exp\big( -\frac{(x_{0,I}-x_{0,J})^2}{2c_x^2} -\frac{(x_{1,I}-x_{1,J})^2}{2c_x^2}\big)\\ C_y(I,J) &= exp\big( -\frac{(y_{0,I}-y_{0,J})^2}{2c_y^2} -\frac{(y_{1,I}-y_{1,J})^2}{2c_y^2}\big). \end{aligned}\]
That is, competition strength is evaluated on the basis of their morphology and habitat preferences. For individuals with identical preferences, \(C(I,J)=1\), that is competition is the strongest.
For individual \(I\), the effective number of competitors is \[N_e(I)=\sum_J C(I,J).\]
The probability that an individual survives to the age of reproduction is \[\label{BH} w(I)=\frac{1}{1+(b-1)\frac{N_e(I)^\theta}{K_e(I)}},\] where the parameter \(\theta\) controls the strength of crowding, and \(b>0\) is a parameter (average number of offspring per female; see below). Equation ([BH]) with \(\theta = 1\) is the Beverton-Holt model which represents a discrete-time analog of the logistic model. If \(N_e(I) \ll K_e(I)\), then \(w(I)=1\) (survival probability is one), and if \(N_e(I)=K_e(I)\), then \(w(I)=1/b\).
The relative probability of mating between a female with traits \(f\) and \(c\) and a male with trait \(m\) is \[\label{pref} \psi(f,c\mid m) = \left\{ \begin{array}{ll} \exp \left( -(2c-1)^2 \frac{(f-m)^2}{2\sigma_a^2}\right) & \mbox{if}\ c > 0.5,\\ 1 & \mbox{if}\ c=0.5,\\ \exp \left( -(2c-1)^2 \frac{(f-(1-m))^2}{2\sigma_a^2}\right) & \mbox{if}\ c<0.5, \end{array} \right.\] where parameter \(\sigma_a\) scales the strength of female mating preferences. Under this parameterization, females with \(c=0.5\) mate randomly, females with \(c>0.5\) prefer males whose trait \(m\) is close to the female’s trait \(f\) (positive assortative mating), and females with \(c<0.5\) prefer males whose trait \(m\) is close to \(1-f\) (negative assortative mating).
The pseudo probability that female \(I = \langle f,c,\ldots \rangle\) mates with male \(I^{\prime}= \langle m,\ldots \rangle\) is \[P(I,I^{\prime}) = \psi(I,I^{\prime})\ C_y(I,J) \\\] This is the preference of \(I\) for \(I^{\prime}\), multiplied by their correlation (with respect to habitat preferences \(C_y(I,J)\)). These “pseudo probabilities” are scaled to sum to one (by summing over the available males \(I^\prime\)).
Each mating results in a number of offspring drawn from a Poisson distribution with parameter \(b\). We assume that all adult females mate. This assumption implies that any costs of mate choice, which can easily prevent divergence and speciation, are absent. This assumption also means that the effective population size is increased relative to the actual number of adults.
With probability \(m>0\), each offspring becomes a “migrant.” Each migrant goes to one of the 8 neighboring patches. For patches at the boundary, the probability \(m\) is reduced according to the number of neighbors they have.
There are two options. (1) viability selection \(\rightarrow\) dispersal \(\rightarrow\) mating and offspring production, and (2) dispersal \(\rightarrow\) viability selection \(\rightarrow\) mating and offspring production.
So, we consider the second option only.
\(K_0\) offsprings populate a patch in the upper left corner. All individuals are identical homozygotes such that all traits are exactly in the middle of their range (i.e., \(x_i=y_i=0.5\) for all \(i\)). These initial conditions correspond to the colonization of a system by a small number of genetically similar, poorly adapted generalists.