Whence Lotka-Volterra?

Abstract

Competition in ecology is often modeled in terms of direct, negative effects of one individual on another. An example is logistic growth, modeling the effects of intraspecific competition, while the Lotka-Volterra equations for competition extend this to systems of multiple species, with varying strengths of intra- and interspecific competition. These equations are a classic and well-used staple of quantitative ecology, providing a framework to understand species interactions, species coexistence, and community assembly. They can be derived from an assumption of random mixing of organisms, and an outcome of each interaction that removes one or more individuals. However, this framing is somewhat unsatisfactory, and ecologists may prefer to think of phenomenological equations for competition as deriving from competition for a set of resources required for growth, which in turn may undergo their own complex dynamics. While it is intuitive that these frameworks are connected, and the connection is well-understood near to equilibria, here, we ask the question: when can consumer dynamics alone become an exact description of a full system of consumers and resources? We identify that consumer-resource systems with this property must have some kind of redundancy in the original description, or equivalently there is one or more conservation laws for quantities that do not change with time. Such systems are known in mathematics as integrable systems. We suggest that integrability in consumer-resource dynamics can only arise in cases where each species in an assemblage requires a distinct and unique combination of resources, and even in these cases, it is not clear that the resulting dynamics will lead to Lotka-Volterra competition.

Appendices

Appendix A: The Hamiltonian formalism for consumer-resource dynamics

A.1 Competition for a single resource

The construction in the main text for logistic growth (Abiotic resource and separation of timescales) also forms part of a more general picture. Equation 11 is a Hamiltonian system (Arnol’d 2013; Marsden and Ratiu 2013), which can be seen by recasting in terms of a pair of canonical variables, say x(t) and y(t) and a Hamiltonian, H(x, y). For a Hamiltonian system, the defining equations must take the form:

$$\begin{array}{@{}rcl@{}} \frac{dx}{dt} &= &-\frac{\partial H}{\partial y}\\ \frac{dy}{dt}& =& \frac{\partial H}{\partial x}. \end{array} $$

(21)

To see this structure explicitly in our case, there is considerable freedom in choosing the variables x(t) and y(t). An example is

$$\begin{array}{@{}rcl@{}} x= \ln\left( R-\frac{d}{\beta}\right)\\ y= \frac{1}{\beta}\ln(N). \end{array} $$

(22)

The Hamiltonian is then proportional to the conserved quantity we already saw:

$$\begin{array}{@{}rcl@{}} H(x,y) = R(x,y)+ N(x,y) = \frac{d}{\beta}+e^{x} + e^{\beta y}, \end{array} $$

(23)

and in terms of these new variables, we can rewrite the consumer-resource dynamics explicitly as Hamiltonian dynamics:

$$\begin{array}{@{}rcl@{}} \frac{dx}{dt} &=& -\beta e^{\beta y} = -\frac{\partial H}{\partial y}\\ \frac{dy}{dt}& =& e^{x} = \frac{\partial H}{\partial x}. \end{array} $$

(24)

We could also change to another set of canonical variables where the dynamics look even simpler:

$$\begin{array}{@{}rcl@{}} J & =& R+N\\ a & =& \frac{\ln{\frac{R-\frac{d}{\beta}}{N}}}{\beta(R-\frac{d}{\beta}+N)}. \end{array} $$

(25)

With the Hamiltonian defined as before, H(J, a) = J and hence is independent of the new canonical variable a. This leads to very simple equations of motion

$$\begin{array}{@{}rcl@{}} \frac{dJ}{dt} & =& \frac{\partial H(J,a)}{\partial a} = 0 \\ \frac{da}{dt} & =& -\frac{\partial H(J,a)}{\partial J} = -1. \end{array} $$

(26)

The advantage of these variables is that the solution a = −t + const is straightforward to obtain, and this can of course be translated back into the familiar logistic model solution for N(t) if necessary.

While the existence of a conserved quantity is on its own sufficient to derive logistic growth for the consumer, making the Hamiltonian formulation slightly superfluous for that specific purpose, we speculate that the structure of Hamiltonian systems may lead to additional insights, beyond this case. This Hamiltonian picture is already known to generalize to a broad range of ecological dynamics (Plank 1995). In particular, any two-variable system of the form

$$\begin{array}{@{}rcl@{}} \frac{dR}{dt} &= R(b_{1}+b_{11}R +b_{12} N) \\ \frac{dN}{dt} &= N(b_{2} + b_{22}N+b_{21}R). \end{array} $$

(27)

has a conserved quantity and a Hamiltonian formulation provided that b₁₁b₂(b₁₂ − b₂₂) + b₂₂b₁(b₂₁ − b₁₁) = 0 (Plank 1995). Our example above (with a redefinition of R to R − d/β) clearly falls into this category, and another familiar example is the Lotka-Volterra predator-prey equations, for biotic (but unregulated) consumers and resources. This has b₁₁ = b₂₂ = 0, and the Hamiltonian

$$\begin{array}{@{}rcl@{}} H_{pp}= b_{1}\ln{N}+b_{12}N-b_{2}\ln{R}-b_{21}R. \end{array} $$

(28)

On the other hand, not every Hamiltonian will lead to a single-valued expression for N in terms of R. For example, if b₂₁, b₁₂ < 0, and b₂, b₁ > 0 in Eq. 28, we cannot rearrange the equation to extract a single-valued solution for R in terms of N and (the constant) H_pp. And so while the existence of a conserved quantity seems essential for finding an exact description of consumer-resource dynamics in terms of consumers alone, it is not necessarily straightforward (or even possible) to use this quantity to derive an ordinary differential equation for N alone.

A.2 Competition for substitutable resources

Like the single consumer-single resource system, Eq. 16 in the main text can also be recast as a Hamiltonian system, with the Hamiltonian:

$$\begin{array}{@{}rcl@{}} H = R_{1} +R_{2} + N_{1} +N_{2}. \end{array} $$

(29)

To see the Hamiltonian property explicitly, consider (for example) the change of variables:

$$\begin{array}{@{}rcl@{}} x_{1} & = &\ln{R_{1}}\\ x_{2} & =& \ln{R_{2}}\\ y_{1} & = & \frac{1}{\beta_{22}\beta_{11}-\beta_{12}\beta_{21}}\left( \beta_{22}\ln{N_{1}}-\beta_{21}\ln{N_{2}}\right)\\ y_{2} & =& \frac{1}{\beta_{22}\beta_{11}-\beta_{12}\beta_{21}}\left( \beta_{11}\ln{N_{2}}-\beta_{12}\ln{N_{1}}\right). \end{array} $$

(30)

Then, with $H(x_{1},y_{1},x_{2},y_{2}) = e^{x_{1}} + e^{x_{2}} +e^{\beta _{11}y_{1}+\beta _{21}y_{2}}+e^{\beta _{22}y_{2}+\beta _{12}y_{1}}$ defined as above,

$$\begin{array}{@{}rcl@{}} \frac{dx_{1}}{dt} & =&-\frac{\partial H}{\partial y_{1}} = -\beta_{11}e^{\beta_{11}y_{1}+\beta_{21}y_{2}}-\beta_{12}e^{\beta_{22}y_{2}+\beta_{12}y_{1}} \\ \frac{dx_{2}}{dt} & =&-\frac{\partial H}{\partial y_{2}} =-\beta_{21}e^{\beta_{11}y_{1}+\beta_{21}y_{2}}-\beta_{22}e^{\beta_{22}y_{2}+\beta_{12}y_{1}}\\ \frac{dy_{1}}{dt} & =&\frac{\partial H}{\partial x_{1}} = e^{x_{1}}\\ \frac{dy_{2}}{dt} & =&\frac{\partial H}{\partial x_{2}} = e^{x_{2}} \end{array} $$

(31)

So far, the analysis is very similar to the case of one consumer and one resource. On the other hand, there is no second conserved quantity here, and so it is only possible to eliminate one of the four dynamical variables, not two. Finally, we note that in the case of essential resources considered in “Essential abiotic resources and multiplicative colimitation”, we are unable to find a Hamiltonian formalism.

Appendix B: Multiplicative and Liebig colimitation

In the main text, we motivate a particular form of colimitation by multiple resources, where each resource must be gathered (in a prescribed quantity) at the same time in order for the consumer to add biomass. The resulting equations are essentially the same as chemical reaction equations for chemicals in a vat. We now present a sketch of how both the Liebig form (a single limiting reaction) and multiplicative colimitation can arise from limits of the same sequence of reactions. Consider a consumer, C, who reaches an intermediate state, D, after uptake of one unit of resource 1, but where this intermediate state can decay back to state C. This is then followed by uptake of resource 2 to make two consumers (i.e., to generate new biomass):

$$\begin{array}{@{}rcl@{}} C + \text{Res}_{1} \rightleftarrows D\\ D + \text{Res}_{2} \rightarrow 2C. \end{array} $$

(32)

We can represent this system by the ODEs for consumer density N, intermediate consumer density M, and resource concentrations R₁ and R₂:

$$\begin{array}{@{}rcl@{}} \frac{dN}{dt} & =& -\gamma NR_{1} + 2\delta MR_{2}+\mu M \end{array} $$

(33)

$$\begin{array}{@{}rcl@{}} \frac{dM}{dt} & =& \gamma NR_{1} - \delta MR_{2} -\mu M, \end{array} $$

(34)

where I am supposing that the resource concentrations are fixed externally or slowly changing, the γ, δ are various reaction rates and μ is a decay rate of the intermediate state back to C. Next, suppose that we can assume a separation of timescales (precisely what I’ve argued against in general in the main text, but like those cases, there are limits in which this approximation is reasonable). Then, we can assume $\frac {dM}{dt}= 0$ and set

$$\begin{array}{@{}rcl@{}} M =\frac{\gamma NR_{1}}{\mu+\delta R_{2}}. \end{array} $$

(35)

Substituting into Eq. 33, we have that

$$\begin{array}{@{}rcl@{}} \frac{dN}{dt} \simeq -\gamma NR_{1} +\frac{\gamma NR_{1}}{\mu+\delta R_{2}}\left( 2\delta R_{2}+\mu\right). \end{array} $$

(36)

If the decay reaction is rare or absent so that we can ignore μ, then this becomes

$$\begin{array}{@{}rcl@{}} \frac{dN}{dt} & \simeq& -\gamma NR_{1} +\frac{\gamma NR_{1}}{\delta R_{2}}\left( 2\delta R_{2}\right)\\ & = &\gamma N R_{1}. \end{array} $$

(37)

I.e., this limit is consistent with one of the two resources (Res₁) being rate limiting, and the growth rate of the consumer only depends on this resource. On the other hand, we could also assume that μ is very large. In the limit of small $\frac {\delta R_{2}}{\mu }$,

$$\begin{array}{@{}rcl@{}} \frac{dN}{dt} & \simeq& -\gamma NR_{1} +\gamma NR_{1}\left( 1+\frac{\delta R_{2}}{\mu}+\dots\right)\\ & =& \frac{\gamma\delta R_{2}R_{1}N}{\mu} \end{array} $$

(38)

and so we recover the multiplicative form. This is simply saying that if decay via μ overwhelms the formation of the intermediate state, D, then in effect, the consumer still needs to gather both resources simultaneously. Finally, if we assume a separation of timescales but now assume that the first reaction is fast, then we can approximately set $\frac {dN}{dt} = 0$ and obtain

$$\begin{array}{@{}rcl@{}} N = M\frac{2\delta R_{2}+\mu}{\gamma R_{1}}. \end{array} $$

(39)

Substituting this into the remaining equation, we have

$$\begin{array}{@{}rcl@{}} \frac{dM}{dt} & \simeq& M\left( 2\delta R_{2}+\mu - \delta R_{2} -\mu \right)\\ & =& \delta R_{2} M. \end{array} $$

(40)

Hence, both the intermediate state, D, and by Eq. 39 the consumer, C, grow exponentially with rate determined by δR₂. I.e., the second resource is limiting, rather than the first. Therefore, with the caveats necessary to trust this “separation of timescales” approach, all three cases (both of Liebig’s possible laws of the minimum and multiplicative colimitation) can arise as limiting cases of this simple sequence of reactions. It is unclear whether either approximation dominates in colimitation in nature, though there seems to be stronger evidence for Liebig. Perhaps a more systematic approximation method for microbial metabolic networks may pay dividends in understanding which of these cases (if any) apply in a given natural system.