Competition for resources: complicated dynamics in the simple Tilman model
 Joost H. J. van Opheusden^{1}Email author,
 Lia Hemerik^{1},
 Mieke van Opheusden^{2} and
 Wopke van der Werf^{2}
Received: 22 May 2015
Accepted: 14 August 2015
Published: 4 September 2015
Abstract
Graphical analysis and computer simulations have become the preferred tools to present Tilman’s model of resource competition to new generations of ecologists. To really understand the full dynamic behaviour, a more rigorous mathematical analysis is required. We show that just a basic stability analysis is insufficient to describe the relevant dynamics of this deceptively simple model. To investigate realistic invasion and succession processes, not only the stable state is relevant, but also the time scales at which the system moves away from the unstable situation. We argue that the relative stability of saddle points is more important for the actual observed transient dynamics in realistic systems than the predicted asymptotic behaviour towards the stable equilibria. For the mathematical analysis this implies that not only the signs, but also the magnitudes of the eigenvalues of the Jacobi matrix at the stationary points, the rates at which the system evolves, must be considered. We present the underlying mathematics of the Tilman model in a way that should be accessible to any ecologist with a basic mathematical background.
Keywords
Background
If the development of an ecosystem is driven by competition, why doesn’t a single species outcompetes all others and becomes the dominant, if not the only surviving one? Why don’t we always have food chains and food towers instead of food webs and food pyramids? Ecological models that describe the resource competition between different species help to understand biodiversity (Levin 2012). Models that show how and why in some situations several species can coexist, while in slightly different situations one may prevail over others, can explain observations of invasion and succession processes in real life environments. Even very simple models can exhibit such complicated behaviour, but unfortunately also the mathematics behind such simple models can become complicated (EdelsteinKeshet 2005). Graphical analysis tools and numerical simulations do enhance accessibility of these models, but cannot replace the in depth insight obtained from mathematical analysis. In fact, for a full understanding of the dynamics of the transition of a system from one situation to another, mathematical analysis is indispensable. Here, we summarize the elementary mathematics underlying a suite of simple models of resource competition (Tilman 1982), in which the dynamics of consumers and resources are explicitly represented. The original papers presenting the mathematical roots of the model require mathematical skills that most ecologists do not possess. Our aim is to make the presentation as accessible as possible. Thus, we keep mathematical sophistication to the minimum needed to explain how the ecology of the model derives from the mathematics.
In the model as introduced by Tilman there are consumers, plants or animals that are part of a given ecosystem, and resources, other plants or animals, but also water, minerals or light that the consumers need for their maintenance and growth. Initially the model was applied to competition between several species of algae (Tilman 1977), which may explain some specific choices made in the model, also when applied more generally. The state variables are the overall densities of consumers and resources within an ecosystem. The model describes the development of these densities in time. There is no direct interaction between the consumers, nor between the resources. Only in the interaction between the consumer and the various resources can there be a combined effect of the different resource densities. The dynamics of the densities is described mathematically by a system of coupled ordinary differential equations.
In “The Tilman model” we discuss the general form of the equations for multiple species competing for several resources, and critically discuss assumptions made in the model. In “Stability analysis of the Tilman model” the model is simplified, c.f. specified, to make it amenable for mathematical analysis, and we discuss what further biological implications are associated with the restricted model. It will turn out that to a large extent the model can be solved quite generically, without full specification of the interaction. We develop a stability analysis for a system of up to two different species competing for two resources, and recapitulate the earlier derived conditions for a stable coexistence. The conditions are mathematical relations between model parameters. Moreover, the analysis yields the relevant time scales at which the model operates, not only for the stable asymptotic state, but also for the unstable, possibly transient states.
To investigate the full dynamic behaviour of the models, which we do in “Numerical model calculations”, we will need to specify the interactions within the model further, and with those the stability results. For all models we investigate how a relatively unpopulated system develops towards the final state with one or two consumers coexisting with the resources. It turns out that indeed all stationary states are important for the overall dynamics, and also the rates, or inversely the time scales, at which the system moves away from the unstable states. Finally we review the biological assumptions made, and explain how we can understand the observed model behaviour in terms of its biological background, or, vice versa, how the model helps to better understand the ecological behaviour of the biological system.
The Tilman model
The dynamical equation for the resources in (1), like that for the consumers, contains a growth and a decline term. The growth rate g _{ j } of any given resource depends only on the resource density itself. Within this paper we will describe the net inflow of nutrients with the familiar chemostat model \(g_{j} \left( {R_{j} } \right) = a_{j} \left( {s_{j}  R_{j} } \right)\). In the absence of consumers it describes restricted exponential growth of the resource density towards s _{ j }, the stable resource density, at a rate a _{ j }. The a _{ j } and s _{ j } are constants. In fact a is the dilution rate of the chemostat, and 1/a is the average residence time of the nutrient inside the system, defining a time scale of the process. A chemostat model may be appropriate in an experimental setup, and applies for instance to a lake with abiotic nutrients delivered and removed through inflowing and outflowing streams. Some aspects of the model, however, may be somewhat counterintuitive. We will mention these later briefly, and discuss them more extensively in a separate paper.
Consumers use the resources for growth. This leads to the decline term in the equation for the resources. The total rate at which a given resource is consumed is the sum of the consumption rates of all consumers using that particular resource. The consumption rate of a resource by a consumer is proportional to the growth rate of that consumer, with a proportionality factor accounting for the conversion from resource to consumer. In the original model as proposed by Tilman the conversion factor q _{ ji } can be a function of all resource densities, but not of the consumer densities. We will use constant positive conversion factors q _{ ji } for resource R _{ j } used by consumer N _{ i }. This implies that in order to produce a single consumer unit, a fixed amount of each resource is removed.
Further attributes of the ecological system can enter the model through the growth functions \(f_{i} \left( {R_{ 1} , R_{ 2} , \ldots } \right)\). In our general stability analysis we assume the growth rate of a consumer to increase with increasing resource density. A larger availability of a resource will make life easier for the consumer, and a smaller investment in finding the resource is expressed in an increased number of siblings or a decreased probability of starvation. For the model the net effect is the same. Secondly we assume that there can be no growth in the absence of resources; a population cannot grow without a resource being present, but instead will die from want. Within the chemostat model for nutrient supply one must be cautious though, because a low nutrient concentration is associated with a high net nutrient inflow. If consumption is very effective, a low abundance can be combined with a large flow. In reality often consumers need to get hold of the resource in order to benefit from it. If that includes foraging for the resource, it needs to be present in some finite abundance, otherwise the investment in foraging behaviour does not exceed the gain in acquiring the resource. The holy grail is to find the perfect balance between simplicity and applicability of the model (Hilborn and Mangel 1997). We will take the growth rate of the consumers to be zero at zero resource abundance.
For the terminology and a short overview of the mathematical analysis method the reader is referred to “Appendix A: Dynamical systems”. In a stability analysis we first find the stationary or equilibrium states of the system of equations by solving the dynamical equations (1) for the case of zero net growth; all derivatives are identically zero. Note that we use the terms equilibrium and stationary state interchangeably. Instead of state we will also use the term point, as explained in “Appendix A”. Exactly at the equilibrium state all derivatives are zero, nothing changes and the systems is fully stationary. The question is what happens if the system is slightly perturbed, does it move back to its equilibrium, or do the perturbations grow, rendering the stationary state unstable? Linearization of the nonlinear system of equations about the stationary state next yields the Jacobi matrix, the eigenvalues and eigenvectors of which matrix exactly tell us how the model system reacts to small perturbations near the stationary state. We present these calculations in much detail in “Appendix B: Mathematical details of the stability analysis”, so the reader may check the steps along the way, and reproduce those steps in case of a slightly modified model system. “Stability analysis of the Tilman model” will focus on the results of these calculations. Next to the stable equilibria or stationary states, we also investigate the unstable stationary states of the system, as these will turn out to play an important role in the transient dynamics of the system, how it develops towards the stable state. Finally we investigate this full dynamics of the examples treated in the next section for a specific class of growth functions. As the differential equations are nonlinear, we use an approximating numerical solution procedure in “Numerical model calculations”.
Stability analysis of the Tilman model
A single species consuming a single resource
Here f _{ B }(R(t)) is the resource dependent relative growth rate of B, and m _{ B } is its mortality. We perform the stability analysis with a general function f, because we want to show that for the chosen characteristics of this function the stability of the steady state is the same, irrespective of the exact form of f. In the absence of B the resource R is depleted at a rate a _{ R }, while it is replenished to a stable level s _{ R }. When B is present, the resource is additionally depleted at the same rate at which B grows, times a conversion factor q _{ RB }. For this simple system we will do the mathematical derivation in full here, derivations for all systems we consider are given in “Appendix B”. We first determine the stationary states (or points) of the system and their stability properties.
In this case the resource concentration is such that growth exactly compensates for the loss term of the consumer, the net growth is zero. Whether there is such a solution depends on the details of f _{ B }(R). The growth function we have chosen is maximal for infinite resource density. If this maximum growth rate lies below the mortality rate of the consumer, then f _{ B }(R) < m _{ B } for all values of R, and there is no solution of (6). In that case the trivial equilibrium (0, s _{ R }) is a stable node. In fact for any positive value of the resource and consumer density the system will develop towards the trivial equilibrium.
If the coexistence point is biologically relevant, both eigenvalues have a negative real part, and the equilibrium is a stable node or a stable vortex. Only if the stable resource level is more than sufficient for the consumer to overcome its inherent mortality, there is a stable finite size population. In “One consumer and one resource” we investigate the full dynamics for a Holling type II growth function. Further discussion of the ecological implications of these results is deferred to the “Conclusions and discussion”.
Two species competing for a single resource
Stationary points (14) and (16) are only biologically relevant if \(s_{R} > R_{A}^{*}\) or \(s_{R} > R_{B}^{*}\) respectively, that is, when the stable resource level exceeds the required level for net population growth of the consumer species.
The next question is about the stability of the three equilibria as a function of the stable resource level s _{ R }. We assume that \(R_{A}^{*} < R_{B}^{*}\). After all, the names are just conventional. If \(s_{R} > R_{A}^{*}\) (and hence also \(s_{R} > R_{B}^{*}\)), there is not enough resource to sustain any consumer. Both (14) and (16) then are unphysical, if they exist at all, because a density cannot be negative. The only real equilibrium is the trivial one, which is a stable node (see “Appendix B: Two species competing for a single resource”). The next case is when \(R_{A}^{*} < s_{R} < R_{B}^{*}\), so there is enough resource for A to grow to its stable level A*, which now is a positive number. Equilibrium (16) still is unphysical. The trivial equilibrium (12) is a saddle point, and equilibrium (14) is a stable node or a stable vortex. Finally we can have \(R_{A}^{*} < R_{B}^{*} < s_{R}\), in which case all three equilibria are biologically relevant. The trivial equilibrium then is a saddle. Equilibrium (14) still is a stable node or a stable vortex, and (16) is a saddle point.
There seem to be only two possibilities. If the steady nutrient supply is insufficient to sustain either consumers, both species become extinct and the resource reaches its stable level; the system goes to the trivial equilibrium. The alternative is that the species with the lower food requirement reaches a stable level, the other becomes extinct and one of the border equilibria is reached. Close to the equilibrium point this is true, but we cannot draw global conclusions from this analysis. We do not have any information about the actual dynamics of the system away from the stationary points. Numerical investigation for specific parameter values and growth functions, as we will do in “Numerical model calculations: two consumers and one resource”, will show this global behaviour in detail, and will show that in fact unstable points can be extremely relevant for what happens in real systems. The conclusion for now is that there can be coexistence between one consumer and one resource, but a single resource in the long run is insufficient, within the model we study, to sustain two different consumers in stable coexistence.
A single species consuming two resources
Note that the conversion factors q _{ PB } and q _{ RB } can be different for the two resources. Also the rates a _{ P } and a _{ R } can be different for the two resources, in case of a chemostat with a single supply reservoir the dilution rates usually will be the same. The growth rate of the species depends on both abundances, allowing for a tradeoff, at least that seems the case. If both resources are needed, \(f_{B} \left( {P, R} \right)\) will be zero in the absence of either resource.
As before, we will call this the trivial equilibrium. The point with both resources at their stable levels is called the supply point.
Whether (19) has solutions depends on the growth function. We must be a bit more specific now we have two resources. We expect a higher abundance of either resource to give a higher growth potential. Hence we assume a monotonically nondecreasing function \(f_{B} \left( {P, R} \right)\), so \(\partial f/\partial P \ge 0\) and \(\partial f/\partial R \ge 0\) regardless of the abundances of the resources. Those assumptions still leave open all kinds of interactions, like substitutability or synergy between the resources in the consumption pattern, but rules out inhibition, where a high abundance of one or both resources reduces the growth.
The solution of (19) is a contour line of \(f_{B} \left( {P, R} \right)\) in the PRplane at the value m _{ B }, called the zero growth isocline of B. If \(f_{B} \left( {P, R} \right) < m_{B}\) for all abundances, there is no solution. Otherwise, in general, the zero growth isocline gives infinitely many combinations of resource abundances for which (17a) yields a stationary B population size. In the case there are no solutions to (19), the trivial equilibrium is the only stationary point. If there are infinitely many solutions to (19) a second criterion comes from the fact that the same stable Bdensity should satisfy both (17b) and (17c).
As before, we will call this the coexistence point.
In “Appendix B: A single species consuming two different resources” we show that if there is a biologically relevant coexistence point the trivial equilibrium is a saddle point, otherwise it is a stable node. If the amount of resources made available is insufficient to compensate for the mortality, the species will become extinct. It looks like the introduction of a second resource does not add to the complexity of the biological system, it only complicates the mathematics. Having a second resource available does not provide the consumer with an option to exchange between the two, its behaviour is fixed by how the growth function depends on the two resource densities and the fixed values of the other parameters.
Two species competing for two resources
If there are more intersection points satisfying (31) there can be additional equilibria of the same type. The inequalities (35) and (36) are related to the usual graphical analysis of the Tilman model in the PRplane (Tilman 1980; also see Ballyk and Wolkowicz 2011, for a detailed description of a slightly different approach). For the case that the a’s are the same, a common choice for a chemostat, both inequalities state that the supply point lies in the wedge between the lines through (P*, R*), with slopes given by the ratios of the conversion factors. The direction vectors of these lines are called the consumption vectors.
The Qmatrix is already familiar. It plays a role in determining the biological relevance of the coexistence point. The derivatives in the second matrix are evaluated at the intersection point of the null isoclines. The columns are the gradients of the growth functions, which vectors are perpendicular to the contour lines. The determinant indicates how these contour lines cross. If the determinants have opposite sign, we have a saddle point, otherwise it is a stable node or stable vortex.
Numerical model calculations
For a specific model we can readily calculate the full solution of the dynamic equations numerically. A standard forward Euler approximation (Press et al. 1986) with a sufficiently small time integration step will usually perform well. We have used a rather straightforward implementation of the model in Excel. We again consider the four different situations, single consumer with single resource, two consumers and a single resource, single consumer and two resources and two consumers with two resources. For each of the systems we calculate the stationary points and their stability properties as a function of the stable level of the resource(s). Next we investigate the global behaviour by following the trajectory of the system in an appropriate part of phase space by numerically integrating the full nonlinear system of differential equations for an appropriately chosen initial situation.
One consumer and one resource
The half saturation constant k _{ RB } mainly sets the scale for s _{ R }, and hence the resource density R. Similarly q _{ RB } sets the relative scale of the consumer density B in Eq. (2b). The mortality rate m _{ B } sets a time scale for the consumer dynamics, while a _{ R } does the same for the resource. We take k _{ RB } = 1. If the coexistence point is biologically relevant, the trivial equilibrium at zero consumer density is a saddle point, otherwise it is a stable node.
If the timescale for the resource replenishment is chosen substantially larger than that of the consumer mortality, i.e. slow replenishment, the coexistence point is a stable vortex. For m _{ B } = 1, a _{ R } = 0.1, s _{ R } = 1, and q _{ RB } = 0.1, the two stationary points have exactly the same density values as above, but the dynamics is different. The initial values are also the same: R(0) = 0, B(0) = 0.01. Again in the phase plot (Fig. 2d) the system moves towards the trivial equilibrium, lingers there until it speeds up in the unstable direction, but then spirals into the coexistence point. Note the difference in time scale with the previous situation in the time series (Fig. 2c). Once the consumer density starts growing, the resource density drops, but both overshoot their stable value. The relaxation towards the coexistence point shows oscillatory behaviour, and is also slower than in the previous case, but the reduction factor is not as high as that for the a _{ R }. The eigenvalues form a complex pair with negative real part Re(λ) = −0.065. The difference with the λ _{1} of the previous case is less than a factor of 5.
Two consumers and one resource
The two cases of interest are R _{ A } ^{*} < s _{ R } < R _{ B } ^{*} , and R _{ A } ^{*} < R _{ B } ^{*} < s _{ R }. For the resource we take a _{ R } = 1, and we look what the dynamics of the system is as a function of the stable level s _{ R }. The scale factors in both growth functions we take unity, the only difference between the consumers is in the maximal value of the growth function. For A we take m _{ A } = 1, q _{ RA } = 1, \(k_{RA} {\kern 1pt} = 1,\) and f _{ mA } = 3, so R _{ A } ^{*} = 0.5, while for B we take m _{ B } = 1, q _{ RB } = 1, \(k_{RB} {\kern 1pt} = 1\), and f _{ mB } = 2, so R _{ B } ^{*} = 1. A has the advantage, as will be confirmed shortly. For initial state we take R(0) = 0, A(0) = 0.001, and B(0) = 1, so we investigate whether indeed A takes over from B.
Again the devil is in the details. In any case consumer A eventually takes over from B, but it depends on the specific parameter values at which rate the system moves away from the trivial saddle or the unstable equilibrium for B. For instance if the advantage of A over B is substantially less than in the example in Fig. 3b above, for all practical purposes the saddle may appear to be stable, simply because the (numerical or real) experiment does not last long enough. Moreover, if the difference between the two consumers is relatively small, also the takeover away from the stationary points is very slow.
One consumer and two resources
There are only two stationary points, the trivial one where the consumer is absent, and the coexistence point where there is a finite consumer population and both resources are present. If the stable level of either resource falls below the minimal required level, the system moves towards the trivial equilibrium, otherwise it moves towards the coexistence point. For the consumer we set \(m_{B} {\kern 1pt} = 1\), \(q_{PB} {\kern 1pt} = 1\), \(q_{RB} {\kern 1pt} = 1\), and f _{ mB } = 3, for the resources we set a _{ P } = 1, a _{ R } = 1, \(k_{PB} {\kern 1pt} = 1\), \(k_{PR} {\kern 1pt} = 1\), s _{ R } = 1, and we investigate the behaviour of the system as a function of s _{ P }. The minimal required resource levels according to the above parameter values are P _{0} = 0.5 and R _{0} = 0.5, so the stable level for R is sufficient. We start at P(0) = 0, R(0) = 0.
If we take all a’s and q’s one tenth of the value in the first calculation, all stationary points have the same values, only the eigenvalues become complex, which implies oscillating graphs in the time plot (Fig. 4c) and in the phase plot (Fig. 4d) a trajectory initially heading for the trivial equilibrium and eventually spiralling into the stable coexistence point. Note that only the BRplane is plotted, in the time plot it is clear the P and R are fully in phase.
Two consumers and two resources
Parameter values for the resources are a _{ P } = 1, a _{ R } = 1, k _{ PA } = 0.9, k _{ PB } = 0.7, k _{ RA } = 0.8, and k _{ RB } = 1, so P _{ A } = 0.45, P _{ B } = 0.35, R _{ A } = 0.4, and R _{ B } = 0.5. For the consumers the parameters are m _{ A } = 1, q _{ PA } = 1, q _{ RA } = 0.8, f _{ mA } = 3, and m _{ B } = 1, q _{ PB } = 0.8, q _{ RB } = 1, f _{ mB } = 3. In a second series we interchange the q’s. We study the behaviour of the system as a function of the stable resource levels s _{ P } and s _{ R }.
The same analysis but with A(0) = 0 (Fig. 5c, d) shows that the system now eventually moves to the saddle point of consumer B, which in the subspace in which we now move does establish a stable node at B _{ B } ^{*} = 0.5, P _{ B } ^{*} = 0.6 and R _{ B } ^{*} = R _{ B } = 0.5. Indeed three of the eigenvalues of this point are negative, all corresponding eigenvectors have zero for the Acomponent. Note that the latter part of the trajectory in the phase plot (Fig. 5d) moves parallel to the consumption vector of B, as for the case of one consumer and two resources.
If instead we set B(0) = 0 (Fig. 5e, f) the system moves even closer to the trivial equilibrium, because we start with just a tiny amount of A. Once this consumer density starts growing exponentially, the system rapidly moves towards the saddle point of A, with A _{ A } ^{*} = 0.55, P _{ A } ^{*} = P _{ A } = 0.45 and R _{ A } ^{*} = 0.56. Note that now the latter part of the trajectory (Fig. 5f) is parallel to the consumption vector of A.
Once again the importance of the saddle points is exemplified by the majority of the above results. In all cases the system first moves along the stable direction of a saddle point towards it, only to move away after it has switched to the unstable direction. Depending on the detailed values of the parameters this may take arbitrarily long. For each dynamical system there are trajectories that travel about its phase space on a detour around the saddle points before finally ending up in a stable equilibrium.
Conclusions and discussion
The conclusions from these mathematical analyses and simulations are quite generic. For all systems investigated the stationary points are either saddle points or stable nodes or vortices. Moreover there always is at least one stable equilibrium. For the resource that is not surprising, that feature is introduced explicitly in the model. For the consumers it is the result of the interaction. If the available amount of food is sufficient to overcome the inherent mortality of the consumer, it will increase in numbers. That is quite trivial and introduced manually into the model too. What is not so trivial is that an increase in consumer density does not necessarily lead to such a decrease of the food supply that it falls below the required abundance, setting off a chain of events that eventually leads to extinction of the consumer or an infinite repetition of events. A scenario like this is feasible, but in the current model it does not happen. Instead the system converges to a stable coexistence point, like in Fig. 2d or Fig. 4d where the trajectory spirals into the stable vortex.
Another generic feature of the models is that if there are biologically relevant stationary points (either internal or boundary equilibria for the consumers) at least one of these is a stable point. Again this may not be very surprising, but there are also physically relevant stationary points that are unstable. In such cases we always have saddle points, there are no unstable nodes or vortices. That the saddle points are important is shown in the numerical section. In all cases where we start with an almost empty system it first develops towards a saddle point, and lingers there for a considerable time before eventually moving away from it, heading for a stable equilibrium, or yet another saddle point. Of course the actual dynamics depend on the initial condition, but the main message is that whenever the system comes near a saddle point, it may stay there for any length of time, depending on actually how unstable the local equilibrium is. Nonlinear systems are known to possess also other type of equilibria, such as (quasi)periodic, homoclinic or heteroclinic orbits or strange attractors in chaotic systems. The systems as discussed here have none of these, though the orbit along the saddle points may be seen as a sort of precursor of a heteroclinic orbit, with the difference that it ends in a stable equilibrium instead of closing upon itself. Saddle points are abundant in complex systems, and they have a significant impact on the dynamics of such systems. The present ones are not very complex but they are no exception. There is nothing exotic about saddle points, they exist for any choice of parameters, the only issue is whether or not the system comes near them. If the instability of a stationary point is the result of a change in one or more of the model parameters, reflecting a change in system properties, such a system may be arbitrary close to a saddle point. The systems we study also are fully deterministic, there is no stochasticity involved, and all parameters are fully constant. In practice it may not be possible to distinguish whether an observed ecosystem is close to a stable equilibrium, or to a saddle point. In fact, with ever changing external factors and the many degrees of freedom in real systems, the concept of a stable equilibrium may not be very relevant, and a spectrum of eigenvalues that indicates how rapidly a given situation may destabilise, is much more useful. It is surprising that such simple equations as the ones discussed here already show the onset of complicated system behaviour, like in Fig. 3b where the unstable coexistence between consumer B and the resource pertains for quite a while before the more successful invader A takes over. Coexistence of two consumers depending on just a single resource is possible within the Tilman model, as is survival of a population which is being invaded by a more efficient consumer, be it for a finite period. Eventually the model system does develop towards a stable equilibrium, the relevant question is about the rate at which the crossover occurs.
In fact we know much more, from the eigenvalues and eigenvectors of the Jacobi matrix we know exactly how and how fast this stable equilibrium is approached if the parameters are such that 0 < R* < s _{ R }. Depending on the available experimental data we can use this to estimate these parameters and decide upon the goodness of fit whether the model provides a satisfactory description, or whether it should be modified. Alternatively we can accept the model and investigate for instance how the equilibrium behaves as a function of the half saturation constant k _{ RB }, keeping all other parameters constant. The maximal value of the consumer density mathematically is reached for a zero value of this parameter, and the resource density is zero too. Ecologically this seems a pathology, but as explained in the introduction of the model we could have expected this. By virtue of the autonomous dynamics of the resource within the model the resource production is maximized at zero resource density. That it really is a pathology is clear when we realise that for k _{ RB } = 0 the growth function (42) of the consumer is constant, independent of the resource density. This violates the condition that the growth function should be zero for zero resource density. For any finite but (very) small value of k _{ RB } the situation in fact is normal. The equilibrium resource density increases linearly with k _{ RB }, and the consumer density decreases linearly, until it becomes zero when R* = s _{ R }. The ecological rationale is that if the consumer really needs only a small amount of resource to be present to overcome its inherent mortality, it can take maximal profit from it. The growth function and the conversion factor reflect the details of the foraging for and digestion of the resource, but the theory does not give us the exact relation. Additional modelling would be needed.
It helps of course to reduce the number of model parameters, but in fact we are dealing with two quite different things. As we have seen before, a very low resource density leads to a very large resource production, allowing for a very large resource uptake. Even if we ignore the aspects of the conversion factor as we have just mentioned, it is wrong to assume a proportionality between the resource uptake and density. Of course it is possible to modify the model to allow for a choice on the part of the consumer, by introducing just one of the two and having the ratio determined by an optimisation procedure of a separate model for the tradeoff between the resources, yet to be specified. Instead of reducing the number of parameters, this will likely lead to an increase of the number of parameters in an extended model with optimal foraging.
The case of two consumers and two resources proves to be a very complex system, given the simplicity of the equations. This should be no surprise. Indeed, systems with as little as three coupled nonlinear ordinary differential equations can show chaotic behaviour with strange attractors. So four such equations could have given even more fireworks, but apparently these don’t. Actually it has been shown (Huisman and Weissing 1999) that three consumers with three resources can show chaotic behaviour, which might suggest that the equations for the resources don’t add to the complexity. In fact chaotic 3D systems can be of the LotkaVolterra direct competition type, and the whole purpose of the Tilman model is to provide a more indirect specification of the consumer interactions, through the resources. A LotkaVolterralike 2D consumer system can show similar behaviour as the 4D Tilman system. Whether the Tilman models really have the advantage of connecting more directly to ecological systems than equivalent LotkaVolterra system cannot be answered on the base of this investigation. We did argue that the additional parameters introduced by the extra equations for the resource still take on effective values when compared to experiments.
Although the equilibria and their stability are explained in the majority of ecological text books, the transient dynamics and how different time scales are involved in these are not. With this paper we contribute to the understanding of the dynamics of competing organisms in a nonequilibrium context. Due to changing environments more often than not systems are not in equilibrium, and the transitional dynamics is more relevant than the equilibrium one.
List of symbols
 A :

Consumer density [L^{−3}]
 a _{ j } :

Growth factor for resource j (either P or R) [T^{−1}]
 B :

Consumer density [L^{−3}]
 c _{ n } :

Coefficient of the characteristic polynomial (of the n × n Jacobi matrix) [T^{−n }]
 F :

Flow [L^{3}T^{−1}]
 f _{ i }():

Growth function for consumer i (either A or B) [T^{−1}]
 f _{ mi } :

Maximum of the growth function for consumer i [T^{−1}]
 J():

Jacobi matrix [T^{−1}]
 k _{ ji } :

Half saturation constant of resource j in growth function of consumer i [L^{−3}]
 λ:

Eigenvalue (of the Jacobi matrix) [T^{−1}]
 m _{ i } :

Mortality rate of consumer i [T^{−1}]
 N _{ i } :

Density of consumer i [L^{−3}]
 P :

Resource density [L^{−3}]
 q _{ ji } :

Conversion factor from resource j to consumer i [0]
 r :

Resource density [L^{−3}]
 R :

Resource density [L^{−3}]
 R _{ j } :

Density of resource j [L^{−3}]
 s _{ j } :

Stable level for resource j [L^{−3}]
 t :

Time [T]
 V :

Volume [L^{3}]
Declarations
Authors’ contributions
All authors contributed significantly. All authors read and approved the final manuscript.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
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