The emergence of heterotrophy in an eco-evolutionary model: modelling trophic transitions in a resource-based framework with naturally-bounded trait distributions

Abstract

A plankton eco-evolutionary model with an alga that has the metabolic pathways to allow it to function as an autotroph or heterotroph is considered. Ecological constraints dictate that the traits that describe the feeding preferences and abilities of the alga naturally have bounded distributions. The trait distributions are then non-normal, and evolve with the population as it changes its trophic behaviour from an autotroph to a heterotroph. A key result of the simulations is that the populations remain in ecological stasis for many generations while the trait mean slowly adapts—only at the conclusion of this transition does herbivory emerge. After initially adapting to improve its competitive performance as an autotroph, the adapting population eventually emerges as a heterotroph having maximised its share of the resources at the expense of its prey, previously its competitor.

Appendices

Appendix 1: Equilibrium points

The full eco-evolutionary model (1) has eco-evolutionary coexistence equilibrium points $\{x_1^*,x_2^*,a_{21}^*,\rho ^*\}$, that are potentially stable and are the focus of our investigation. One coexistence equilibrium point ${\textit{EP}}_1$ is:

$$\begin{aligned} x_1^*&= \frac{r_1a_{22}-r_2a_{12}-m_1(r_2+a_{22})+m_2(r_1+a_{12})}{r_1a_{22}-r_2a_{12}+a_{11}(r_2+a_{22})}, \\ x_2^*&= \frac{r_1(1-x_1^*)-m_1-a_{11}x_1^*}{r_1+a_{12}}, \\ a_{21}^*&> -a_{12}, \\ \rho ^*&= 1. \end{aligned}$$

(13)

For the parameter values used in the simulations (Table 1) this point lies inside the feasible ecospace $E\equiv \{0<x_1<1,0<x_2<1;0<x_1+x_2<1\}$, and is relevant to the early eco-evolutionary dynamics of the system. Note that when $\rho ^*=1$ the value of $a_{21}^*$ is not fixed as it does not appear in the model Eqs. (1) or (13). Substituting $\rho ^*=1$ into the Jacobian matrix [Eq. (9)] of the model [Eq. (1)] and solving for the eigenvalues gives:

$$\begin{aligned} 2\lambda _{1,2}&= -(r_1+a_{11})x_1^*-(r_2+a_{22})x_2^* \\&\quad \pm \sqrt{{\{(r_1+a_{11})x_1^*-(r_2+a_{22})x_2^*\}^2 +4r_2(r_1+a_{12})x_1^*x_2^*}}, \\ \lambda _3&= 0, \\ \lambda _4&= -\frac{h_2^2}{\beta _2^2}(r_2N^*+a_{21}^*x_1^*). \end{aligned}$$

(14)

As the value of $a_{21}^*>-a_{12}$ is not fixed for ${\textit{EP}}_1$, in the four dimensional eco-evo phase space this equilibrium point in E becomes a half-line of equilibrium points. Further, $\lambda _4$ can change sign and its stability along the $\rho$ manifold can change, and is negative when $a_{21}^*$ is sufficiently positive, that is when $a_{21}^*>-r_2N^*/x_1^*$.

The other coexistence equilibrium point ${\textit{EP}}_0$ is:

$$\begin{aligned} x_1^*&= \frac{r_1(m_2+a_{22})-m_1a_{22}+m_2a_{12}}{a_{11}a_{22}-a_{12}a_{21}^*-r_1(a_{21}^*-a_{22})}, \\ x_2^*&= \frac{r_1(1-x_1^*)-m_1-a_{11}x_1^*}{r_1+a_{12}}, \\ a_{21}^*&= -a_{12}, \\ \rho ^*&= 0. \end{aligned}$$

(15)

The eigenvalues of this point are:

$$\begin{aligned} 2\lambda _{1,2}&= -(r_1+a_{11})x_1^*-a_{22}x_2^* \pm \sqrt{{\{(r_1+a_{11})x_1^*-a_{22}x_2^*\}^2 +4r_2(r_1+a_{12})x_1^*x_2^*}}, \\ \lambda _3&= - \frac{h_1^2}{\beta _1} x_1^*, \\ \lambda _4&= \frac{h_2^2}{\beta _2^2}(r_2N^*-a_{12}x_1^*). \end{aligned}$$

(16)

The eigenvalues $\lambda _{1,2}$ on the $\{x_1,x_2\}$ manifold remain the same, and are negative for all values used in the numerical simulation. The eigenvalue along the $a_{21}$ manifold $\lambda _3$ is always negative for positive values of $x_1$, and the eigenvalue along the $\rho$ manifold $\lambda _4$ is also negative as $r_2N^*<a_{12}x_1^*$. This point is then stable. Note that the fourth eigenvalues $\lambda _4$ of ${\textit{EP}}_1$ and ${\textit{EP}}_0$ have opposite signs.

Table 1 Parameter values used in Eq. (1) to draw Figs. 1 and 2

Appendix 2: Parameter values

Although this work is conceptually based on the results of Bell (2012), the ecological model is heuristic, with population interactions represented in their simplest form, so we have not attempted to reproduce the timescale of evolution noted by Bell. The parameter values have been chosen to produce interesting, representative dynamics in a reasonable computational time. We do not report a parameter sensitivity analysis here, but note that the evolutionary time scales are sensitive to both the evolutionary and ecological parameters. In particular, we note that the $x_2$ density-independent mortality parameter $m_2$ has a significant effect on the evolutionary timescale, supporting the view that adaptation is driven by the rate at which less-fit individuals are removed from the population.