Framework for analyzing ecological trait-based models in multidimensional niche spaces

We develop a theoretical framework for analyzing ecological models with a multidimensional niche space. Our approach relies on the fact that ecological niches are described by sequences of symbols, which allows us to include multiple phenotypic traits. Ecological drivers, such as competitive exclusion, are modeled by introducing the Hamming distance between two sequences. We show that a suitable transform diagonalizes the community interaction matrix of these models, making it possible to predict the conditions for niche differentiation and, close to the instability onset, the asymptotically long time population distributions of niches. We exemplify our method using the Lotka-Volterra equations with an exponential competition kernel.

Figure 1

(Main figure) The spectrum of $\mathcal{J},\left\{J_K\right\}$ (red dots), is calculated with Eq. (17) and shown as a function of the increasing wavelength $K$, after a reordering as explained in Sec. 4a. The blue line has been added for clarity. Parameter values: $L=4,{\Delta }_l=2,R=1$, and $\sigma =2$. (Inset) The components of eigenmode $v^{\left(11\right)}$ for the same parameter values.

Figure 2

(Main figure) The concentrations $X_I$ as a function of the individuals $I$ (represented as binary sequences at the top of the figure), obtained by running stochastic simulations for time $t=V*{10}^2$ with $V={10}^3$ and other parameter values as in Fig. 1. (Inset) The absolute value of $A\left(K\right)$ [Eq. (20)] obtained from the final individual distribution shown in the main figure.

Figure 3

The spectrum of $\mathcal{J},\left\{J_K\right\}$ (red dots), is calculated with Eq. (17) and shown as a function of $K$ for parameter values: $L=3,{\Delta }_1=3,{\Delta }_2=9$, and ${\Delta }_3=2$ with $\sigma =2$ and $R=1$. The blue line has been added for clarity. Note that the (stable) eigenvalues corresponding to $K=1,19,39$ are not visualized in the plotted range.

Figure 4

Final niche distribution of a single run. Parameter values as in Fig. 3 but with $V={10}^2$. The simulation has run for time $T={10}^{3}V$.

Figure 5

The average absolute value of $A\left(K\right)$ (red dots), obtained by averaging formula (20) over ${10}^4$ runs. Parameter values as in Fig. 3 but with $V={10}^2$. Each simulation has run for time $T={10}^{3}V$. The blue line has been added for clarity.