Geometry of ecological coexistence and niche differentiation

A fundamental problem in ecology is to understand how competition shapes biodiversity and species coexistence. Historically, one important approach for addressing this question has been to analyze consumer resource models using geometric arguments. This has led to broadly applicable principles such as Tilman's $R^{*}$ and species coexistence cones. Here, we extend these arguments by constructing a geometric framework for understanding species coexistence based on convex polytopes in the space of consumer preferences. We show how the geometry of consumer preferences can be used to predict species which may coexist and enumerate ecologically stable steady states and transitions between them. Collectively, these results provide a framework for understanding the role of species traits within niche theory.

Figure 1

MacArthur consumer resource model. (a) A species $i$ consumes resource $\alpha$ with preference set by consumer preferences $c_{i\alpha }$. (b) Population dynamics for a system with two resources and four species described by Eq. (1) (see the Appendix pp1-s1 for parameters).

Figure 2

Zero net growth isoclines (ZNGIs) and coexistence cones for ecosystems with $S=5$ species and $M=2$ resources. (a) ZNGIs for each species are plotted in the space of resource abundances. The highlighted region corresponds to the polytope defining the infeasible region where the growth rate is negative for all species. (Inset) ZNGI for one species where different growth-rate regions are labeled. (b) Trajectories of solutions in resource phase space (black lines), coexistence cones (green cones), and species abundances $N_i$ as a function of time for five different choices of carrying capacity indicated by ${\mathbf{K}}_A,\cdots ,{\mathbf{K}}_E$. Dynamics in Fig. (1) correspond to trajectory with carrying capacity ${\mathbf{K}}_D$. See interactive demonstrations in Appendix pp1-s4.

Figure 3

Relationship between coexistence cones and the positive convex hull (PCH) formed by rescaled consumption vectors. Left: Coexistence cones are highlighted in relation to the ZNGIs for competitive species. If the vector of carrying capacities, $\mathbf{K}$, belongs to cone $\alpha$, the species $i=1,i=5$ can coexist; if $\mathbf{K}$ belongs to cone $\beta$, then species $i=1,i=4$ can coexist; otherwise, only $i=5,i=1,i=4$ can survive separately. Right: The positive convex hull (PCH) of rescaled consumption vectors ${\mathbf{C}}_i$ has faces that are dual to the arrangement of the coexistence cones. Each possible ecologically stable combination of surviving species is represented by a face of the PCH.

Figure 4

Positive convex hull (PCH) formed by rescaled consumption vectors for an ecosystem with $M=3$ resources and $S=20$ species. In each panel, the left plot shows the infeasible region polytope formed by the ZNGI, the resource supply vector $\mathbf{K}$, and the associated coexistence cone. The top plot shows the species populations $N_i$ as a function of time. The right plot depicts the PCH; the face corresponding to the steady state for the choice of $\mathbf{K}$ is indicated by large green dots. Small dots indicate consumption vectors, ${\mathbf{C}}_i$, that lie in the interior of the PCH and correspond to species that go extinct for all choices of $\mathbf{K}$. The directed graph shows possible transitions between these steady states as $\mathbf{K}$ is varied when no reinvasion occurs. A value of $\mathbf{K}$ is chosen arbitrarily in each pane such the species which survive demonstrate each face of the PCH. The panes are organized to reflect the face lattice of the PCH. (Also see the interactive Mathematica notebook.)

Figure 5

Enumerating steady states and transitions using the PCH. All possible steady states and allowed transitions (without reinvasion) for the ecosystem in Fig. 4 can be enumerated by considering the geometry of the faces on the PCH.

Figure 6

Positive convex hull (PCH) of rescaled consumer preferences predicts coexistence properties when species fitness is modified. Panels show PCH (top) and species abundances as a function of time for fixed resource supply vector $\mathbf{K}$ (bottom) as the fitness of species 2 is increased by decreasing the parameter $m_2$ for the ecosystem analyzed in Figs. 2 and (3). Increasing the fitness of species 2 interrupts the coexistence of species 1 and 5, followed by different levels of competitive exclusion.

Figure 7

Positive convex hull (PCH) construction for a CRM with a nonlinear growth rate. Each row corresponds to a value of ${\mathbf{R}}^{★}$; the left panels show $\mathbf{K},{\mathbf{R}}^{★}$, ZNGIs, coexistence cones; the right panels show the PCH construction and surviving species for the given ${\mathbf{R}}^{★}$. For a nonlinear growth rate, $g_i\left(\mathbf{R}\right)$, the PCH's form depends on ${\mathbf{R}}^{★}$. This figure corresponds to the model from Eqs. (14) and (15).

Figure 8

Construction of the positive convex hull of consumption vectors with three resources. Upper left: the positive convex hull of rescaled consumption vectors, represented by large blue dots. The projection of the rescaled consumption vectors onto the coordinate planes and axes are displayed with differently colored smaller dots. Different view perspectives labeled A, B, C are shown with small eye cartoons. Upper right, lower left, lower right: The PCH along with the projected rescaled consumption vectors are shown from the different labeled view perspectives, emphasizing that the construction of the PCH is preserved under projection onto coordinate planes.

Figure 9

Relationship between coexistence cones, the PCH formed by rescaled consumption vectors, and stably coexisting species in the consumer resource model with Externally supplied resources (eCRM). Left: Coexistence cones and ZNGIs are highlighted in the resource-abundance phase space. Notice that the coexistence cones have the same structure but are in different positions compared to Fig. (3) Center: PCH formed by rescaled consumption vectors; the faces of this PCH enumerate stably coexisting species. (cf., Fig. 3). Right: Population dynamics of all sets of stably coexisting species; each plot corresponds to a different choice of supply rates, ${\kappa }_{\alpha }$.

Figure 10

Positive convex hull (PCH) formed by rescaled consumption vectors for an ecosystem with $M=3$ resources with dynamics given by the consumer resource model with externally supplied resources (eCRM). The structure and layout of this figure is identical to that of Fig. 4, but the dynamics and coexistence cones shown are those of the eCRM model [Eq. (7)].

Figure 11

Convex polytope enumerating coexisting species for the Tilman consumer resource model. Left: resource-abundance phase space, with $\mathbf{K}, {\mathbf{R}}^{★}$, and the trajectory $\mathbf{R}\left(t\right)$ plotted as in Fig. 2; here, however, negative values of resource abundances are shown. The species population dynamics are additionally inset. Right: The relevant convex polytope which is the convex hull of ${\left\{{\mathbf{C}}_i\right\}}_{i=1}^S\cup \left\{\mathbf{0}\right\}$ has faces which enumerate the possible sets of species that can coexist. The face corresponding to the coexisting species shown in the left figure is highlighted in green.