Quantifying the Likelihood of Co-existence for Communities with Asymmetric Competition

Abstract

Trade-offs in performance of different ecological functions within a species are commonly offered as an explanation for co-existence in natural communities. Single trade-offs between competitive ability and other life history traits have been shown to support a large number of species, as a result of strong competitive asymmetry. We consider a single competition-fecundity trade-off in a homogeneous environment, and examine the effect of the form of asymmetry on the likelihood of species co-existing. We find conditions that allow co-existence of two species for a general competition function, and show that (1) two species can only co-exist if the competition function is sufficiently steep when the species are similar; (2) when competition is determined by a linear function, no more than two species can co-exist; (3) when the competition between two individuals is determined by a discontinuous step function, this single trade-off can support an arbitrarily large number of species. Further, we show analytically that as the degree of asymmetry in competition increases, the probability of a given number of species co-existing also increases, but note that even in the most favourable conditions, large numbers of species co-existing along a single trade-off is highly unlikely. On this basis, we suggest it is unlikely that single trade-offs are able to support high levels of bio-diversity without interacting other processes.

Appendices

Appendix A

We note that since c(z)+c(−z)=2 constant, we have that c(−z)−1=1−c(z)>0 for positive z. Therefore, we can subtract (12) from (11) to get

$$\frac{(1-z-c(z))-((1-z)c(-z)-1)}{c(-z)-1}=\frac{z(c(-z)-1)}{(c(-z)-1)}=z. $$

Therefore, the upper and lower bounds for x ₂ coincide at z=0, and for z>0 the upper bound is always greater than the lower, i.e. there is always a region where co-existence is possible. Therefore, if the upper bound (11) is non-increasing, both it and the lower bound (12) never exceed the value they achieve at z=0, and there is only a region of co-existence in the positive quadrant when the limit

$$\lim_{z \to0}\frac{1-z-c(z)}{1-c(z)} >0. $$

As both numerator and denominator tend to zero, we use l’Hopital’s rule to get that this limit is given by

$$\frac{-1-c'(0)}{-c'(0)}, $$

which is positive for c′(0)<−1.

It remains to demonstrate that (11) is non-increasing for positive z. The derivative of the function is given by

$$\frac{c(z)-1-zc'(z)}{(1-c(z))^2}. $$

If c′(z)=0, then this becomes

$$\frac{-1}{1-c(z)}<0, $$

since c(z)<1 for all z>0. When c′(z)<0, then we note that the function is non-increasing for

$$z\leq\frac{c(z)-1}{c'(z)}. $$

The derivative of the right-hand side is given by

$$1-\frac{c''(z)(c(z)-1)}{c'(z)^2}, $$

which is greater than one for all positive z. Therefore, this point increases faster than z. Noting that the derivative of (11) is zero when z=0, we can therefore conclude that for all z>0, this upper bound is indeed non-increasing.

Appendix B

Without losing any generality, we can order traits such that x ₁>x ₂>⋯>x _n so that since p is decreasing, p(x ₁)<p(x ₂)<⋯<p(x _n ). When ϵ=0, Eqs. (14) become

$$\frac{d N_i}{d\tau}=N_i \Biggl(p(x_i)-\sum _{j=1}^n N_{j} \Biggr), \quad i=1, \ldots,n. $$

Thus, if i<k,

$$\frac{d}{d\tau} \log(N_i/N_k) = p(x_i)-p(x_k)<0. $$

This shows that for i<k

$$N_i(\tau) = N_k(\tau) e^{( p(x_i)-p(x_k)) \tau} \rightarrow0, \quad \tau\rightarrow\infty, $$

since N(t) is bounded. This shows that N _i (τ)→0 as τ→∞ for i=1,2,…,n−1. It is intuitive that the remaining species density N _n (τ)→p(x _n ) as τ→∞. This can be shown by first noting that the equation for the dynamic of N _n reduces to the time-dependent logistic equation:

$$ \dot{N}_n = p(x_n) N_n \biggl(1- \frac{N_n}{K(t)} \biggr), $$

(19)

where the time-dependent carrying capacity is given by

$$K(t) = p(x_n)e^{p(x_n)t}\biggr/\Biggl(\sum _{i=1}^n e^{p(x_i)t}\Biggr). $$

One may verify that the explicit solution to (19) is

$$N_n(t) = \frac{N_n(0) e^{p(x_n)t}}{1+N_n(0)\sum_{i=1}^n ( \frac {e^{p(x_i)t}-1}{p(x_i)} )}\rightarrow p(x_n)\quad\mbox{as } t\rightarrow\infty. $$

We have thus shown that the species with minimum trait 0 will send any other species to extinction.

Appendix C

Lemma 1

Whenever an interior fixed point exists for the n-species model given by (14), it is both unique and globally asymptotically stable relative to the interior of the region of space where all N _i are positive and, therefore, the system displays permanence.

Proof

It is well known that when (1) admits an interior steady state, it is globally stable (also known as LV-stable) if the matrix −C=((−c _ij (x))) is dissipative, that is, there exists a positive diagonal matrix D such that the real symmetric matrix D(−C)+(−C)^T D is negative definite and, therefore, has only negative eigenvalues (e.g. Hofbauer and Sigmund 1998). For the model with discontinuous competition, the competition matrix takes the form (in the region x ₁>x ₂>⋯>x _n ) given by Eq. (22). Let D be a positive diagonal matrix with diagonal entries θ _i (i=1,…,n). Then (1), with competition matrix C as in (22), admits a fixed point that globally attracts interior trajectories whenever the real symmetric matrix A=(DC+C ^T D), given by

has all positive eigenvalues, which is the case if all the leading principal minors are positive.

We have fixed θ _i >0 positive, so we restrict ourselves to looking at m×m leading principal minors of the matrix A with m≥2:

As the determinant of a matrix is not changed when one row or column is subtracted from another, we then subtract the (m−1)th column from the mth column, and the ith column from the (i+1)th column for all 1≤i≤m−1 to obtain the matrix

Removing the common factors in columns 2 up to m give that the determinant has a factor

$$(-1)^{m-1}\prod_{i=1}^{m-1}( \theta_i-\theta_{i+1}) $$

which is then multiplied by

Subtracting row m−1 from row m, and row i from row i+1 for all i<m gives this determinant as

Therefore, the m×m leading principal minor for m>1 is given by

$$ -(2\epsilon)^{m-2} \bigl( (1-\epsilon)^2 \theta_1-(1+\epsilon)^2\theta_m \bigr)\prod _{i=1}^{m-1}(\theta_i- \theta_{i+1}). $$

(20)

It remains to be shown that the θ _i can be chosen such that all leading principal minors (20) are positive, which is the case when ((1−ϵ)² θ _i −(1+ϵ)² θ _j )<0 and (θ _i −θ _j )>0 for i<j. Setting

$$\theta_i=n-\frac{1}{(1+\epsilon)^{n+1-i}}, $$

then we can use j−i≥1 to calculate that

To show that ((1−ϵ)² θ _i −(1+ϵ)² θ _j )<0, we note that

and that for ϵ=0, we have ((1−ϵ)² θ ₁−(1+ϵ)² θ _n )=0. Differentiating this with respect to epsilon gives

for all 0≤ϵ≤1 and n≥2. Therefore, since ϵ=0 is a root for the function, for any ϵ>0, we have

$$\bigl( (1-\epsilon)^2\theta_1-(1+\epsilon)^2\theta_n \bigr)<0 $$

and therefore

$$\bigl( (1-\epsilon)^2\theta_i-(1+\epsilon)^2\theta_j \bigr)<0. $$

Therefore, all the leading principal minors are positive, meaning that A=DC+C ^T D is positive definite. Therefore, D(−C)+(−C)^T D is negative definite, as required for global stability. Global stability immediately implies permanence. □

Appendix D

Here, we calculate the probability of co-existence for an n species version of (14). Since we are using the step function (5), the matrix C(x) is piecewise constant. At the interior fixed point, the bracketed terms are equal to zero, which therefore reduces the model to p(x)=C(x)N, with p the vector of the growth rate of each species and N the vector with ith component N _i . Since C(x) is non-singular, this is the rearranged to give the solution N=C ⁻¹(x)p(x). We are interested in the volume of x-space for which N=C ⁻¹(x)p(x)>0. To find this volume, we find the volume in p-space where p ₁<p ₂<⋯<p _n which satisfies, since then x ₁>x ₂>⋯>x _n and so C(x) is a constant, and non-singular matrix C, so that

$$ C^{-1}\mathbf{p} = \mathbf{0}. $$

(21)

Equation (21) defines a series of planes in p-space that all pass through the origin. When an ordering p _n >p _n−1>⋯>p ₁ is assumed without any loss of generality, these planes form an n dimensional pyramid when intersected with the unit cube. The volume of this pyramid is then the probability of the n species model permitting an interior fixed point and, therefore, the probability of all n species co-existing due to the stability result in Appendix C.

With the ordering p ₁<p ₂<⋯<p _n , equivalent to x ₁>x ₂>⋯>x _n , the competition matrix C takes the form

(22)

It is relatively simple to show that C is non-singular, allowing us to calculate the volume of the n dimensional pyramid. We need to find the n points at which n of the planes intersect the face p _n =1. To find these points, we note that the edges of the n dimensional pyramid must be orthogonal to each of the n−1 planes in p-space defined by (21) that meet at that edge. Since C ⁻¹ C=I _n , each column of C is orthogonal to all bar one of the rows of C ⁻¹ Therefore, the edges of the n dimensional pyramid point in the direction of the columns of C. These edges meet the plane p _n =1 at the non-zero corners of the n dimensional simplex, so it is simple to show that these points are given by the columns of C, scaled such that the value of p _n =1. Therefore, the n non-zero vertices of the simplex that lie in the plane p _n =1 are given by

Theorem 1

The probability of co-existence, is given by

$$ V_n=P(\mathit{coexist}_n)= \begin{cases} \frac{\epsilon^n}{(1+\epsilon)^{n-1}} &\mbox{even }n\geq2, \\ \frac{\epsilon^{n-1}}{(1+\epsilon)^{n-1}} &\mbox{odd }n\geq3. \end{cases} $$

(23)

Proof

The volume of the n dimensional simplex is given by

Since there are n! of these volumes, one for each ordering of the traits, so the total volume is $V_{n} = n! \bar{V}_{n}$. Let $U_{n} = n! (1+\epsilon)^{n-1}\bar{V}_{n}$. Then

We now show that U _n =ϵ ² U _n−2. To see this, first subtract the second row from the first row. Having done this, in the new determinant subtract the first column from the second column. This gives

which gives U _n =ϵ ² U _n−2 as required. Now quick calculations show that U ₂=ϵ ² and U ₃=ϵ ² so that U _n =ϵ ⁿ for n≥2 even and U _n =ϵ ⁿ⁻¹ for n≥3 odd.

 □