Cross-Feeding Dynamics Described by a Series Expansion of the Replicator Equation

Abstract

Understanding how ecosystems evolve and how they respond to external perturbations is critical if we are to predict the effects of human intervention. A particular class of ecosystems whose dynamics are poorly understood are those in which the species are related via cross-feeding. In these ecosystems the metabolic output of one species is being used as a nutrient or energy source by another species. In this paper we derive a mathematical description of cross-feeding dynamics based on the replicator equation. We show that under certain assumptions about the system (e.g., high flow of nutrients and time scale separation), the governing equations reduce to a second-order series expansion of the replicator equation. By analysing the case of two and three species we derive conditions for co-existence and show under which parameter conditions one can expect an increase in mean fitness. Finally, we discuss how the model can be parameterised from experimental data.

Appendices

Appendix A: An Estimate of the Higher-Order Terms

In order for the truncated series in Eq. (8) to be valid, we need to control the tail, i.e. the higher-order terms.

First we need to define the notation of multi-index I. Let I=i ₁ i ₂…i _k where all i _j ∈{1,…,n}, and let x _I be the product $x_{i_{1}} x_{i_{2}} \ldots x_{i_{k}}$ of fractions of species. We will also use the notation Ii for the concatenation i ₁…i _k i. Finally, we denote the length of the multi-index I by |I|.

Definition 1

We say that the cross-feeding process is Level Limited of degree d if there are positive constants c ₀ and c, where c<1 such that for every multi-index I with |I|≥d−1 and every index i,

$$\mathcal{E}_{I i} \leq c_0 c^{|I|}. $$

Note that if the cross-feeding process is monotone in the following way:

$$\frac{\mathcal{E}_{I i}}{ \mathcal{E}_{I}} \leq c < 1 $$

for all multi-indices I with |I|≥d−1, then it is Level Limited of degree d.

Theorem 1

The sum of the higher-order terms of degree d, Ω(d), is bounded above by κ ^d+1/(κ+γ)^d. Furthermore, if the process is level limited of degree d, we have that the sum of the higher-order terms of degree d are bounded in the following way:

$$ \varOmega(d) \leq \frac{c_0 \gamma \kappa^{d+1} c^{d}}{(\kappa + \gamma)^{d+1} (\kappa(1-c) +\gamma)}. $$

(21)

Before we prove this, we need a lemma as a generalisation of Eqs. (6) and (7), which is easily proved using induction.

Lemma 1

For any multi-index I,

$$\widehat{s}_I = \frac{\kappa^{|I|} \gamma x_I}{(\kappa + \gamma)^{|I|+1}}. $$

Let us now prove the proposition starting with Eq. (21).

Proof of Theorem 1

To simplify the notation, let us consider the case where d=2 and recall that η=κ/(κ+γ). We start by noting that the sum over every multi-index I with fixed length p is

By then using the lemma and the assumption that the process is Level Limited of degree d=2, we have that

Now, if we cannot assume that the sequence is Level Limited, we always have the trivial estimate $\mathcal{E}_{I} \leq 1$ for any multi-index I. Repeating the train of inequalities above, we end up with the following estimate:

$$\varOmega(2) = \gamma \sum_{p=2}^\infty \eta^{p+1} \sum_{|I|=p} x_I \mathcal{E}_{I i} \leq \gamma \bigl(\eta^3+ \eta^4+ \cdots \bigr) = \frac{\gamma \eta^3}{1-\eta} = \frac{\kappa^3}{(\kappa+\gamma)^2}. $$

It is now straightforward to extend the proof to any degree d other than two. □

Appendix B: Intransitivity and Permanence for Three Species

In this section we present a conditions for intransitivity and permanence for three interacting species.

Definition 2

We say that a set containing N species is intransitive if the constituent species can be ordered in such a way that the pairwise dynamics between species i and i+1 for i=1,…,N−1 are such that the fixed point (x _i ,x _i+1)=(1,0) is unstable and the fixed point at (x _i ,x _i+1)=(0,1) is stable, and where the fixed point (x _N ,x ₁)=(1,0) is unstable and (x _N ,x ₁)=(0,1) is stable.

Theorem 2

Consider three species involved in cross-feeding described by (1) and (9) and assume that Ω(2)=0. If the inequalities

$$ \mathcal{E}_{i} - \mathcal{E}_{j} > \eta( \mathcal{E}_{jj} - \mathcal{E}_{j i}) $$

(22)

and

$$ \mathcal{E}_{i} - \mathcal{E}_{j} > \eta( \mathcal{E}_{ij} - \mathcal{E}_{ii}) $$

(23)

hold for all pairs of species i and j, then the three species form an intransitive triple. Further, if the inequality

$$ \varGamma_{12} \varGamma_{23} \varGamma_{31} < 1, $$

(24)

where

$$\varGamma_{ij} = \frac{\mathcal{E}_j-\mathcal{E}_i + \eta(\mathcal{E}_{jj}-\mathcal{E}_{ji})}{\mathcal{E}_j - \mathcal{E}_i + \eta(\mathcal{E}_{ji}-\mathcal{E}_{ii})}, $$

holds, then the system exhibits permanence.

Proof

We begin by noting that inequality (22) is equivalent to condition (14), which describes the criteria for the fixed point x=0 being unstable in the two species system, and that the second inequality (23) is the reverse of (15), hence a condition for the fixed point at x=1 being stable. Since this is assumed to hold between all three species, we have, according to the definition, an intransitive set.

For the second part of the theorem, we will make use of the technique described in Sect. 4.3 (Jansen 1987), here extended to the case of three species. First we define the function P(x,y)=x ^a y ^b(1−x−y)^c, where x is the fraction of species 1, y is the fraction of species 2, and the fraction of species 3 is given by 1−x−y. The constant a,b,c are assumed to be real and positive. Now P satisfies

$$\frac{dP(x(t),y(t))}{d t} = P(x,y) \varPsi (x,y) $$

with Ψ(x,y)=aw ₁(x,y)+bw ₂(x,y)+cw ₃(x,y), where $w_{i}(x,y)=\phi_{i}(x,y)-\bar{\phi}(x,y)$. In order to determine the permanence of the system, we need to determine if there exists constants a,b,c>0 such that Ψ(x ^⋆,y ^⋆)>0 for all fixed points (x ^⋆,y ^⋆) on the boundary of the system. Since we know that the system is intransitive, we also know that the fixed points on the boundary are located at the corners of the simplex, i.e. at (x ^⋆,y ^⋆)=(0,0), (0,1) and (1,0). Evaluating Ψ at these points, we get, by using (9),

For notational convenience, we now define

$$\lambda_{ji} = \gamma \eta \mathcal{E}_i+\gamma \eta^2 \mathcal{E}_{ji}. $$

We can now rewrite the above inequality as

$$ \varPsi(0,0) = a (\lambda_{31} - \lambda_{33}) + b (\lambda_{32} - \lambda_{33}) > 0, $$

(25)

and similarly for the two other fixed points, we get

$$ \varPsi(0,1) = a (\lambda_{21} - \lambda_{22}) + c (\lambda_{23} - \lambda_{22}) > 0 $$

(26)

and

$$ \varPsi(1,0) = b (\lambda_{12} - \lambda_{11}) + c (\lambda_{13} - \lambda_{11}) > 0. $$

(27)

We will now try to eliminate the constants a,b,c and determine for which values of the λs these inequalities hold. We proceed by rewriting (25) as

$$ a > b \frac{\lambda_{33}-\lambda_{32}}{\lambda_{31}-\lambda_{33}}, $$

(28)

where we have assumed that λ ₃₁>λ ₃₃. However this assumption corresponds to (22). We now rearrange (27) into

$$ a < c \frac{\lambda_{23}-\lambda_{22}}{\lambda_{22}-\lambda_{21}}, $$

(29)

under the warranted assumption that λ ₂₂>λ ₂₁ i.e. (23). Combining (28) and (29), we obtain

$$b \frac{\lambda_{33}-\lambda_{32}}{\lambda_{31}-\lambda_{33}} < a < c \frac{\lambda_{23}-\lambda_{22}}{\lambda_{22}-\lambda_{21}}. $$

This eliminates a from the system, and we can instead write

$$\frac{c}{b} > \frac{(\lambda_{33}-\lambda_{32})(\lambda_{22}-\lambda_{21})}{(\lambda_{31}-\lambda_{33}) (\lambda_{23}-\lambda_{22})}, $$

where we have assumed that λ ₂₂<λ ₂₃, i.e. (22). Lastly (26) gives us, under the assumption that λ ₁₁>λ ₁₃, i.e. (23),

$$\frac{c}{b} < \frac{\lambda_{12}-\lambda_{11}}{\lambda_{11}-\lambda_{13}}. $$

We now have lower and upper bounds for c/b, and we can conclude that constants a, b, c exist that satisfy the original inequalities if

$$\frac{(\lambda_{33}-\lambda_{32})(\lambda_{22}-\lambda_{21})}{(\lambda_{31}-\lambda_{33}) (\lambda_{23}-\lambda_{22})} < \frac{\lambda_{12}-\lambda_{11}}{\lambda_{11}-\lambda_{13}}, $$

where in the last step we made use of the fact that λ ₁₂>λ ₁₁, i.e. (22). This last inequality is equivalent to (24), and hence we know that the system is permanent in this case. □