7 Competition

Different species frequently compete for limiting resources, and as a result have negative impacts on each other. For example, change in species composition during secondary succession (Fig. 7.1) appears mediated by, among other things, a species’ ability to intercept light, grow, and cast shade over other species. This chapter addresses very simple ways to represent such interactions between species. Unless I specify otherwise, “competition” in this chapter will refer to competition between species, or interspecific competition.

*Changes in abundances of six species of asters during early secondary succession. This turnover of perennial weeds appears mediated by competition for light, among other factors (data from the long-term Buell-Small Succession study [http://www.ecostudies.org/bss/]).*

Figure 7.1: Changes in abundances of six species of asters during early secondary succession. This turnover of perennial weeds appears mediated by competition for light, among other factors (data from the long-term Buell-Small Succession study [http://www.ecostudies.org/bss/]).

In this chapter we introduce direct and indirect interspecific competition (Fig. 7.2). Direct competition is an interaction in which individuals experience a direct negative effect of another individual through aggressive interaction via contact or signaling. Indirect competition are negative interactions mediated by another factor, such as a limiting resource. When grass species compete for nitrate and ammonium, it is the shortage of those nutrients that causes the negative effects, and that shortage results from their consumption by competing species.

Figure 7.2: We can model competition as direct negative effects, or as indirect interactions mediated through a resource.

We start with modeling direct interactions using the Lotka-Volterra framework. We then move on to indirect interactions using a consumer-resource model.

7.1 Lotka-Volterra Interspecific Competition

Our models of density-dependent growth were built upon the assumption that individuals within a single population had negative effects on each other, which we refered to as intraspecific competition. Here we assume that individuals of different species also have negative effects on each other.

The classic model of competitive interactions is the continuous Lotka-Volterra model of interspecific competition (Kingsland 1985). We build directly on the logistic growth model by adding an additional term for negative density dependence, this time, with the competing species. Now we have to label our populations and all of our parameters using subscripts. These subscripts identify the species to which the parameter relates. For instance, $r_1$ is the intrinsic rate of increase for species 1, and $\alpha_{12}$ is the per capita effect of species 2 on 1. \[\begin{align} \frac{d N_1}{d t}&=r_1N_1\left(1-\alpha_{11}N_1-\alpha_{12}N_2\right)\tag{7.1}\\ \frac{d N_2}{d t}&=r_2N_2\left(1-\alpha_{21}N_1-\alpha_{22}N_2\right) \tag{7.2} \end{align}\] You will notice in (7.1) that the denisty-dependence term includes the total effect of species 1 on itself ($\alpha_{11}$) and the total effect of species 2 and species 1 ($\alpha_{12}$) (Table 7.1).

It is useful to understand where these subscripts come from. Why are they read from right to left — why not left to right? As we saw in our models of structured population growth, it comes from the underlying linear algebra used to work with all these equations. We define all of the $\alpha$’s together as a matrix, \[\begin{equation} \tag{7.3} \mathbf{\alpha} = \begin{pmatrix} \alpha_{11} & \alpha_{12} \\ \alpha_{21} & \alpha_{22} \end{pmatrix} = \begin{pmatrix} 0.010 & 0.005\\ 0.008 & 0.010 \end{pmatrix} \end{equation}\] The subscripts on the $\alpha$s represent the row and column of the coefficient; $\alpha_{12}$ is in the first row, second column. This merely reflects how mathematicians describe matrix elements and dimensions — row $\times$ column. When we use matrix multiplication, $\alpha_{12}$ becomes the effect of species 2 (column) on species 1 (row). In this case, $\alpha_{11}=\alpha_{22}=0.01$, $\alpha_{21}=0.008$, and $\alpha_{12}=0.005$. Thus, both species have greater effects on themselves than on each other. Remember, the larger the $\alpha$, the larger the effect.

Table 7.1: Parameters of the continuous 2 species Lotka-Volterra competition model. The rest of the chapter explains the meaning and motivation.
Parameter	Description
$r_i$	Instantaneous rate of increase; intrinsic rate of growth; individuals produced per individual per unit time.
$\alpha_{ii}$	Intraspecific density dependence; intraspecific competition coefficient; the negative effect of an individual of species $i$ on its own growth rate.
$\alpha_{ij}$	Interspecific density dependence; interspecific competition coefficient; the effect of interspecific competition; the negative effect that an individual of species $j$ has on the growth rate of species $i$.
$K_i$	$1/\alpha_{ii}$; carrying capacity of species $i$; the population size obtainable by species $i$ in the absence of species $j$.
$\alpha^\prime_{ij}$	$\alpha_{ij}/\alpha_{ii}$; the relative importance of interspecific competition.
$\beta_{ij}$	$\alpha_{ij}/\alpha_{jj}$; the invasion criterion for species $i$; the relative importance of interspecific competition; the importance of the effect of species $j$ on species $i$ relative to the effect of species $j$ on itself.

These equations are also commonly represented using carrying capacity, $K_i$, to summarize intraspecific effects on abundance, and coefficients to modify the intraspecific effects quantified with $K_i=1/\alpha_{ii}$. This representation looks like \[\begin{align} \frac{d N_1}{d t}&=r_1N_1\left(\frac{K_1-N_1-\alpha'_{12}N_2}{K_1}\right) \tag{7.4}\\ \frac{d N_2}{d t}&=r_2N_2\left(\frac{K_2-N_2-\alpha'_{21}N_1}{K_2}\right). \tag{7.5} \end{align}\] In this version, $K_1=1/\alpha_{11}$, and note that $\alpha'_{12}$ differs from $\alpha_{12}$. The $\alpha'_{12}$ in eq. (7.4) merely modifies the effect of $1/K_1$. It turns out the $\alpha'_{1,2}$ is equal to the ratio of the interspecific and intraspecific per capita effects, or \[\begin{align} \alpha_{12}&=\frac{\alpha'_{12}}{K_1}\notag\\ \alpha'_{12}&=\frac{\alpha_{12}}{\alpha_{11}}. \end{align}\] Another useful measure of the relative importance of interspecific competition is $\beta_{ij}=\alpha_{ij}/\alpha_{jj}$ (see Invasion criteria, below).

7.2 Equilbria

In this section, we describe how we find equilibria in a simple multispecies model, by solving the growth equations for zero, much the way we did in Chapters 3 and 4. We begin with isoclines, and move on to boundary and internal equilibria and invasion criteria.

7.2.1 Isoclines

An isocline is, in general, a line connecting points on a graph or map that have equal value. For instance, a topographic map has elevation isoclines, connecting points of equal elevation. Here, our isoclines will connect points in state space at which the growth rate for species $i$ equals zero — every point on that line will represent $\dot{N}_i=0$. We call these zero net growth isoclines (ZNGI), sometimes called “zingees” by people who prefer an economy of words.

A zero net growth isocline is the set of all points for which the growth of a population is zero, when the population size of a competing species is held constant. A two-species equilibrium is a point at which the growth rates of both populations are zero. We saw previously that the carrying capacity of a single species logistic population is an equilibrium. With two species, it gets a tad trickier.

We find the isocline of a species by setting its growth rate equal to zero and solving the equation for that species in terms of the other species. As an example, find the zero net growth isocline for $N_2$. We find below that there is a straight line that describes all points for which $d N_2/dt=0$, if $N_1$ were held constant.

We start by setting (7.2) to zero, and solving for $N_2$. \[\begin{align*} \frac{d N_2}{d t}&=r_2N_2\left(1-\alpha_{21}N_1-\alpha_{22}N_2\right) \\ 0 &=r_2N_2\left(1-\alpha_{21}N_1-\alpha_{22}N_2\right)\\ 0 &= 1-\alpha_{21}N_1-\alpha_{22}N_2\\ N_2 &=\frac{1}{\alpha_{22}} - \frac{\alpha_{21}}{\alpha_{22}}N_1 .\tag{7.6} \end{align*}\] Recall that the formula for a straight line is $y=mx + b$ where $m$ is the slope and $b$ is the intercept on the $y$ axis. We can see that the expression for $N_2$ in (7.6) is a straight line, where $y=N_2$, $m= \alpha_{21}/\alpha_{22}$, and $b=1/\alpha_{22}$ (Fig. 7.3). When $N_1$ is zero, $N_2=1/\alpha_{22}$. This is precisely what we saw in Chapter 3 (logistic growth), that a single species logistic population has an equilibrium at its carrying capacity, $K=1/\alpha$.

The isocline (Fig. 7.3) shows that as the competitor’s population size, $N_1$, becomes larger, $N_2$ declines by $\alpha_{21}/\alpha_{22}$ for each additional individual of the competing species 1, until finally $N_2=0$ when $N_1=1/\alpha_{21}$.

Graphing an isocline isn’t too hard, so let’s graph something similar to Fig. 7.3. First, we define a new matrix of competition coefficients, where $\alpha_{11}=\alpha_{22} > \alpha_{12}=\alpha_{21}$.

a <- matrix(c(  0.01, 0.005,
                  0.005, 0.01), ncol=2, byrow=TRUE)

We create an expression to plot the $N_2$ isocline, as a function of possible values of $N_1$.

N2iso <- expression(1/a[2,2] - (a[2,1]/a[2,2])*N1)

We then specify $N_1$, and then evaluate and plot $N_2$. We add arrows to remind us of what happens if $N_2$ is above or below the value on the isocline.

N1 <- 0:200
plot(N1, eval(N2iso), type='l',
      ylim=c(0, 200), xlim=c(0, 200),
      ylab=expression("N"[2]))
arrows(x0=90, y0=150, x1=90, y1=80, length=0.1)
arrows(x0=75, y0=0, x1=75, y1=50, length=0.1)

The isocline for $N_2$ (Fig. 7.3) is the line at which $d N_2/d t =0$ for a fixed value of $N_1$. Just as in the single species logistic growth model, if $N_2$ exceeds its equilibrium, it declines, and if $N_2$ is less than its equilibrium, it grows. The isocline (Fig. 7.3 is the set of balance points between positive and negative growth. This is reflected in the arrows in Fig. 7.3 — if the $N_2$ is ever above this isocline, it declines and if it is ever below this isocline, it rises. This isocline shows that whether $N_2$ increases or decreases depends on $N_1$.

*Phase plane plots of each of the two competing species, with Lotka-Volterra zero growth isoclines. Arrows indicate population trajectories. Recall $K = 1/alpha$.*

Figure 7.3: Phase plane plots of each of the two competing species, with Lotka-Volterra zero growth isoclines. Arrows indicate population trajectories. Recall $K = 1/alpha$.

By analogy, the isocline for species 1 turns out to be \[\begin{align} N_1 &=\frac{1}{\alpha_{11}} - \frac{\alpha_{12}}{\alpha_{11}}N_2. \tag{7.7} \end{align}\]

Note that these isoclines are merely equations for straight lines, and it is easy to do nonsensical things, such as specify coefficients that result in negative population sizes. Therefore, let us proceed with some thoughtfulness and care.

7.2.2 Finding equilibria

By themselves, the isoclines tell us that if species 2 becomes extinct ($N_2=0$), then species 1 reaches its carrying capacity ($N_1=1/\alpha_{11}$) (Fig. 7.3). Similarly, if $N_1=0$, then $N_2=1/\alpha_{22}$. These are important equilibria, because they verify the internal consistency of our logical, and they provide end-points on our isoclines.

If the species coexist ($N_1,\,N_2 >0$) it means that they must share one or more points on their isoclines — such an equilibrium is the point where the lines cross. We find these equilibria by solving the isoclines simultaneously. A simple way to do this is to substitute the right hand side of the $N_2$ isocline (7.6) in for $N_2$ in the $N_1$ isocline (7.7). That is, we substitute an isocline of one species in for that species’ abundance in another species’ isocline. Combining eqs. (7.6) and (7.7), we get \[\begin{align} N_1 &=\frac{1}{\alpha_{11}} - \frac{\alpha_{12}}{\alpha_{11}}\left(\frac{1}{\alpha_{22}} - \frac{\alpha_{21}}{\alpha_{22}}N_1 \right)\notag\\ N_1 &=\frac{1}{\alpha_{11}} - \frac{\alpha_{12}}{\alpha_{11}\alpha_{22}}+ \frac{\alpha_{12}\alpha_{21}}{\alpha_{11}\alpha_{22}}N_1 \notag\\ N_1\left(1-\frac{\alpha_{12}\alpha_{21}}{\alpha_{11}\alpha_{22}}\right)&= \frac{\alpha_{22}-\alpha_{12}}{\alpha_{11}\alpha_{22}} \notag\\ N_1^*&=\frac{\alpha_{22}-\alpha_{12}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}} \tag{7.8} \end{align}\] When we do this for $N_2$, we get \[\begin{align} N_2^*&=\frac{\alpha_{11}-\alpha_{21}}{\alpha_{22}\alpha_{11}-\alpha_{12}\alpha_{21}}\tag{7.9} \end{align}\]

We now have the values for $N_1^*$ and $N_2^*$ at the point at which their isoclines cross (Fig. 7.4, upper left). These equilibria apply only when isoclines cross within feasible state space.

The expressions for $N_1^*$ and $N_2^*$ look pretty complicated, but can we use them to discern an intuitive understanding for species 1? First, we see that $r_i$ is not in the expressions for the equilibria — they do not depend on $r_i$ in this two-species model. Second, we can confirm that as interspecific competition intensity falls to zero ($\alpha_{12}=\alpha_{21}=0$), each species reaches its own carrying capacity. That is, when putative competitors occupy sufficiently different niches and no longer compete, then they both reach their own carrying capacities.

We can also say something a little less obvious about species 1. What happens when the negative effect of the competitor, $N_2$, starts to increase, that is, as $\alpha_{12}$ gets bigger? Or, put more obtusely but precisely, let’s find \[\begin{equation} \label{eq:1} N_1^* = \lim_{\alpha_{12} \to \infty}\frac{\alpha_{22}-\alpha_{12}}{\alpha_{22}\alpha_{11}-\alpha_{12}\alpha_{21}} \end{equation}\] that is, find the limit of the equilibrium (7.8) as $\alpha_{12}$ gets very large. Well, the $\alpha_{ii}$ become effectively zero because $\alpha_{12}$ gets so big. This leaves $-\alpha_{12}/(-\alpha_{12}\alpha_{21})=1/\alpha_{21}$. Huh? This means simply that as the negative effect of the competitor increases, the abundance of species 1 becomes increasingly dependent upon $\alpha_{21}$, its negative effect on its competitor. Thus we have an arms race: as the negative effect of its competitor increases, the abundance of a species depends increasingly on its ability to suppress the competitor.

Summarizing, we see that in the absence of interspecific competition, species are attracted toward their carrying capacities. Second, if interspecific competition is intense, then a species’ carrying capacity becomes less important, and its abundance is increasingly determined by its ability to suppress its competitor.

7.2.3 Coexistence — the invasion criterion

Based on the numerators in eqs. (7.8) and (7.9), it seems that $N_1^*$ and $N_2^*$ may both be greater than zero whenever $\alpha_{ii}-\alpha_{ji} >0$. This is, indeed, the case. Below we step through the analysis of what we refer to as the “invasion criterion,” which specifies the conditions for $N_i^* > 0$.

In general, the details of any model and its dynamics may be quite complicated, but as long as we know whether a species will always increase when it is rare, or invade,⁴³ then we know whether it can persist in the face of complex interactions. Thus we don’t need to find its equilibrium, but merely its behavior near zero.

How do we determine whether a species can increase when rare? Let’s explore that with the Lotka-Volterra competition model. We can start letting species 1’s growth equation (7.1) be greater than zero. \[\begin{equation*} \tag{7.10} 0 < r_1N_1\left(1-\alpha_{11}N_1-\alpha_{12}N_2\right). \end{equation*}\] From this we can see that in the absence of any density dependence ($\alpha=0$) and assuming $r_1>0$, and $N_1>0$, the population grows exponentially. Further, we can see that $d N/d t >0$ as long as $\left(1-\alpha_{11}N_1-\alpha_{12}N_2\right)>0$.

To determine whether $N_1$ can increase when rare we want to figure out what happens when $N_1$ gets very close to zero. We start by expressing $d N_1 / d t$ completely in terms of $N_1$ and $\alpha$. We do this by substituting $N_2$’s isocline (7.6) in place of $N_2$ in (7.1). We then solve this for any growth greater than zero. \[\begin{align} 0&< \left(1-\alpha_{11}N_1-\alpha_{12}N_2\right)\notag\\ 0&<\left(1-\alpha_{11}N_1-\alpha_{12}\left(\frac{1}{\alpha_{22}}-\frac{\alpha_{21}}{\alpha_{22}}N_1\right)\right) \tag{7.11} \end{align}\]

Now — what is the value of (7.11) as $N_1 \to 0$? We can substitute 0 for $N_1$, and (7.11) becomes \[\begin{align} 0&<\left(1-\alpha_{12}\left(\frac{1}{\alpha_{22}}\right)\right) \notag\\ 0&<1-\frac{\alpha_{12}}{\alpha_{22}}\notag\\ \alpha_{12}&<{\alpha_{22}}. \end{align}\] What is this saying? It is saying that as long as $\alpha_{12}<\alpha_{22}$, then our focal species can persist, increasing in abundance from near zero — $N_1$ will increase when rare, that is, it will successfully invade.

For two species to both persist, or coexist, it must be that case that \[\begin{equation*} \alpha_{12}<\alpha_{22} \quad , \quad \alpha_{21}<\alpha_{11} . \tag{7.12} \end{equation*}\]

Simply put, for species to coexist stably, their effects on themselves must be greater than their effects on each other (Fig. 7.4, upper left).

7.2.4 Other equilibria

Given our isoclines and equilibria above, what other logical combinations might we see, other than coexistence? Here we list others, and provide graphical interpretations (Fig. 7.4).

Species 1 can invade when rare, but species 2 cannot (Fig. 7.4, lower left) \[\begin{equation*} \alpha_{12} < \alpha_{22} \quad,\quad \alpha_{21}>\alpha_{11} \end{equation*}\] This leads to competitive exclusion by species 1 — species 1 wins. This is referred to as a boundary equilibrium, because it is on the boundary of the state space for one species. Equilibria where all $N_i>0$ are referred to as internal equilibria.

Species 2 can invade when rare, but species 1 cannot (Fig. 7.4, upper right) \[\begin{equation*} \alpha_{12}>\alpha_{22} \quad,\quad \alpha_{21}<\alpha_{11} \end{equation*}\] This leads to competitive exclusion by species 2 — species 2 wins. This is the other boundary equilibrium. Note that for both this and the previous boundary equilibrium, the equilibrium equations, (7.8) & (7.9), can return $N^*$ that are negative or too large ($>K$). Recall that these equations derive from simple equations of straight lines, and do not guarantee that they are used sensibly — equations aren’t dangerous, theoreticians who misuse equations are dangerous.

Neither species can invade when rare (Fig. 7.4, lower right).

\[\begin{equation*} \alpha_{12} > \alpha_{22} \quad,\quad \alpha_{21} > \alpha_{11} \end{equation*}\]

This creates an unstable internal equilibrium—exclusion will occur, but either species could win. This condition is sometimes referred to as founder control (Bolker, Pacala, and Neuhauser 2003) because the identity of the winner depends in part on the starting abundances. It creates a saddle in state space. What the heck is a saddle? More on that below. It suffices to say that from some directions, an saddle attracts the trajectories of the populations, while from other directions, it repels the trajectories.⁴⁴

$*Phase plane diagrams of Lotka-Volterra competitors under different invasion conditions. Horizontal and vertical arrows indicate directions of attraction and repulsion for each population (solid and dased arrows); diagonal arrows indicate combined trajectory. Circles indicate equilibria; additional boundary equilibria can occur whenever one species is zero. Left to right: stable eq., $a_{ji} < a_{ii}$; $N_2$ wins, $a_{12} > a_{22}$; $N_1$ wins, $a_{21} > a_{11}; saddle attractor-repellor, $a_{ji} > a_{ii}$.*$ $*Phase plane diagrams of Lotka-Volterra competitors under different invasion conditions. Horizontal and vertical arrows indicate directions of attraction and repulsion for each population (solid and dased arrows); diagonal arrows indicate combined trajectory. Circles indicate equilibria; additional boundary equilibria can occur whenever one species is zero. Left to right: stable eq., $a_{ji} < a_{ii}$; $N_2$ wins, $a_{12} > a_{22}$; $N_1$ wins, $a_{21} > a_{11}; saddle attractor-repellor, $a_{ji} > a_{ii}$.*$ $*Phase plane diagrams of Lotka-Volterra competitors under different invasion conditions. Horizontal and vertical arrows indicate directions of attraction and repulsion for each population (solid and dased arrows); diagonal arrows indicate combined trajectory. Circles indicate equilibria; additional boundary equilibria can occur whenever one species is zero. Left to right: stable eq., $a_{ji} < a_{ii}$; $N_2$ wins, $a_{12} > a_{22}$; $N_1$ wins, $a_{21} > a_{11}; saddle attractor-repellor, $a_{ji} > a_{ii}$.*$ $*Phase plane diagrams of Lotka-Volterra competitors under different invasion conditions. Horizontal and vertical arrows indicate directions of attraction and repulsion for each population (solid and dased arrows); diagonal arrows indicate combined trajectory. Circles indicate equilibria; additional boundary equilibria can occur whenever one species is zero. Left to right: stable eq., $a_{ji} < a_{ii}$; $N_2$ wins, $a_{12} > a_{22}$; $N_1$ wins, $a_{21} > a_{11}; saddle attractor-repellor, $a_{ji} > a_{ii}$.*$

Figure 7.4: Phase plane diagrams of Lotka-Volterra competitors under different invasion conditions. Horizontal and vertical arrows indicate directions of attraction and repulsion for each population (solid and dased arrows); diagonal arrows indicate combined trajectory. Circles indicate equilibria; additional boundary equilibria can occur whenever one species is zero. Left to right: stable eq., $a_{ji} < a_{ii}$; $N_2$ wins, $a_{12} > a_{22}$; $N_1$ wins, $a_{21} > a_{11}; saddle attractor-repellor, $a_{ji} > a_{ii}$.

7.3 Dynamics at the Equilibria

Here we use eigenanalysis to analyze the properties of the equilibrium, whether they are attractors, repellers, or both, and whether the system oscillates around these equilibria.

For logistic growth, we assessed stability with the partial derivative of the growth rate, with respect to population size. If it was negative the population was stable, and the more negative the value, the shorter the return time. Here we build on this, and present a general recipe for stability analysis (Morin 1999):

Determine the equilibrium abundances of each species by setting its growth equation to zero, and solving for $N$.
Create the Jacobian matrix. This matrix represents the response of each species to changes in its own population and to changes in each other’s populations. The matrix elements are the partial derivatives of each species’ growth rate with respect to each population.
Solve the Jacobian. Substitute the equilibrium abundances into the partial derivatives of the Jacobian matrix to put a real number into each element of the Jacobian matrix.
Use the Jacobian matrix to find the behavior of the system near the equilibria. The trace, determinant, and eigenvalues of the Jacobian can tell us how stable or unstable the system is, and whether and how it cycles.

7.3.1 Determine the equilibria

We just did this above. Given (7.8) and (7.9), we see that the $\alpha$ determine completely $N_1^*$ and $N_2^*$. This is not true for Lotka-Volterra systems with more than two species; such systems also depend on $r_i$.

Finding equilibria in R can be done using expressions for the equilibria, $N_1^*$ and $N_2^*$. An R expression is a symbolic representation of an equation. These will be symbolic representations that we later evaluate using eval().

N1Star <- expression( (a22-a12)/(a22*a11 - a12*a21) )
N2Star <- expression( (a11-a21)/(a22*a11 - a12*a21) )

Next we create the $\mathbf{\alpha}$ and evaluate our expressions.

a11 <- a22 <- 0.01;  a12 <- 0.001; a21 <- 0.001
N1 <- eval(N1Star); N2 <- eval(N2Star)
N1

## [1] 90.90909

7.3.2 Create the Jacobian matrix

The next step is to find each partial derivative. The partial derivatives describe how the growth rate of each species changes with respect to the abundance of each other species and with respect to its own abundance. Thus a positive value indicates that a growth rate increases as another population increases. This might be the case with predator growth rate and its prey population. A negative value indicates a growth rate decreases as another population increases. This is likely to be true for competitors or prey growth rate and its predator population.

Here, we work through an example, deriving the partial derivative of species 1’s growth rate with respect to itself.

First let’s expand the growth rate of species 1 (7.1)⁴⁵ \[\begin{align} \frac{d N_1}{d t}&=\dot{N_1} =r_1N_1 - r_1\alpha_{11}N_1^2 - r_1\alpha_{12}N_2N_1. \end{align}\]

Now we derive the partial differential equation (PDE)⁴⁶ with respect to $N_1$, treating $N_2$ as a constant. \[\begin{equation} \frac{\partial \dot{N_1} }{\partial N_1}=r_1 - 2r_1\alpha_{11}N_1 - r_1\alpha_{12}N_2 \end{equation}\] We should think of this as the per capita effect of species 1 on its growth rate.

To derive the PDE with respect to $N_2$, we treat $N_1$ as a constant, and find \[\begin{equation} \frac{\partial \dot{N_1} }{\partial N_2}= - r_1\alpha_{12}N_1. \end{equation}\] This is the per capita effect of species 2 on species 1’s growth rate.

We then do the same for $\dot{N_2}$, and so derive the full matrix of PDE’s, \[\begin{align} \tag{7.13} \left( \begin {array}{cc} \frac{\partial \dot{N_1}}{\partial N_1}&\frac{\partial \dot{N_1}}{\partial N_2}\\ \frac{\partial \dot{N_2} }{\partial N_1}&\frac{\partial \dot{N_2}}{\partial N_2} \end {array} \right) = \left( \begin {array}{cc} r_1 - 2r_1\alpha_{11}N_1 - r_1\alpha_{12}N_2& - r_1\alpha_{12}N_1\\ - r_2\alpha_{21}N_2&r_2 - 2r_2\alpha_{22}N_2 - r_2\alpha_{21}N_1\\ \end {array} \right). \end{align}\]

This matrix of PDEs is the Jacobian matrix, or simply the “Jacobian.” As differential equations, they describe the slopes of curves (i.e. the slopes of tangents of curves) at a particular point. That is, they describe the straight line interpretations as that point. As partial differential equations, they describe how the growth rates change as population sizes change.

Finding partial differential equations and the Jacobian matrix in R can be done using expressions. Here we create equations or expressions for the for the growth rates, $\dot{N_1}$ and $\dot{N_2}$, and use these to find the partial derivatives. First, expressions for the growth rates:

dN1dt <- expression( r1 * N1 - r1 * a11 * N1^2 - r1 * a12 * N1 * N2 )
dN2dt <- expression(r2*N2 - r2*a22*N2^2 - r2*a21*N1*N2)

Next, we use each expression for $\dot{N}$ to get each the partial derivatives with respect to each population size. Here we use the R function D() (see also ?deriv). We reveal here the result for the first one only, the partial derivative of $\dot{N_1}$ with respect to itself, and then get the others.

ddN1dN1 <-D(dN1dt, "N1")
ddN1dN1

## r1 - r1 * a11 * (2 * N1) - r1 * a12 * N2

Here we find the remaining PDEs.

ddN1dN2 <- D(dN1dt, "N2")
ddN2dN1 <- D(dN2dt, "N1")
ddN2dN2 <- D(dN2dt, "N2")

Last we put these together to create the Jacobian matrix, which is itself an expression that we can evaluate again and again.

J <- expression( matrix(c(eval(ddN1dN1), eval(ddN1dN2),
                     eval(ddN2dN1), eval(ddN2dN2)),
                   nrow=2, byrow=TRUE) )

7.3.3 Solve the Jacobian at an equilibrium

To solve the Jacobian at an equilibrium, we substitute $N_1^*$ (7.8) and $N_2^*$ (7.9) into the Jacobian matrix (7.13). Refer to those equations now. What is the value of $N_1$ in terms of $\alpha_{ii}$ and $\alpha_{ij}$? Take that value and stick it in each element of the Jacobian (7.14). Repeat for $N_2$. When we do this, and rearrange, we get, \[\begin{align} \mathbf{J}= \left( \begin {array}{cc} -r_1\alpha_{11}\left(\frac{\alpha_{22}-\alpha_{12}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}}\right)& - r_1\alpha_{12}\left(\frac{\alpha_{22}-\alpha_{12}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}}\right)\\ - r_2\alpha_{21} \left(\frac{\alpha_{11}-\alpha_{21}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}}\right)& -r_2\alpha_{22}\left(\frac{\alpha_{11}-\alpha_{21}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}}\right)\\ \end {array} \right). \tag{7.14} \end{align}\]

Yikes $\ldots$ seems a little intimidating for such a small number of species. However, it is remarkable how each element can be expressed as a product of $-r_i\alpha_{ij}N_i^*$, where $i$ refers to row, and $j$ refers to column.

Evaluating the Jacobian matrix in R is the easy part, if we defined it as above. Assuming that above we selected particular $\mathbf{\alpha}$, used these to determine $N_1^*$ and $N_2^*$, found the PDEs and created an expression for the Jacobian matrix, and labeled everything appropriately, we can then evaluate the Jacobian at an equilibrium. For $\alpha_{ii}=0.01$ and $\alpha_{ij}=0.001$ (see above) we find

r1 <- r2 <- 1
J1 <- eval(J)
J1

##             [,1]        [,2]
## [1,] -0.90909091 -0.09090909
## [2,] -0.09090909 -0.90909091

Note that all of these PDEs are negative for this equilibrium. This indicates a stable equilibrium, because it means that each population’s growth rate slows in response to an increase in any other.

7.3.4 Use the Jacobian matrix

Just the way we used eigenanalysis to understand long term asymptotic behavior of demographic matrices, we can use eigenanalysis of the Jacobian to assess the long-term asymptotic behavior of these competing Lotka-Volterra populations. We can again focus on its dominant, or leading, eigenvalue ($\lambda_1$). The dominant eigenvalue will be the eigenvalue with the greatest real part, and not necessarily the eigenvalue with the greatest magnitude.⁴⁷ In particular, the dominant eigenvalue, $\lambda_1$, may have a real part for which the magnitude⁴⁸ is smaller, but which is less negative or more positive (e.g., $\lambda_1=-.01$, $\lambda_2=-1.0$). For continuous models, the dominant eigenvalue, $\lambda_1$, is approximately the rate of change of a perturbation, $x$, from an equilibrium, \[\begin{equation} \tag{7.15} x_t=x_0e^{\lambda_{1}t}. \end{equation}\] Thus, the more negative the value, the faster the exponential decline back toward the equilibrium (i.e., toward $x=0$). As we did for logistic growth, we should think of the dominant eigenvalue as a “perturbation growth rate”: negative values mean a decline of the perturbation, and positive values indicate an increase in the perturbation, causing the system to diverge or be repelled away from the equilibrium.

In addition to the dominant eigenvalue, we need to consider the other eigenvalues. Table 7.2 provides a summary for interpreting eigenvalues with respect to the dynamics of the system. The eigenvalues depend upon elements of the Jacobian, and values calculated from the elements, notably the determinant, the trace, and the discriminant; a similar set of rules of system behavior can be based upon these values (Roughgarden 1998). For instance, the Routh-Hurwitz criterion for stability tells us that a two-species equilibrium will be locally stable, only if $\mathbf{J_{11}} + \mathbf{J_{22}} < 0$ and if $\mathbf{J_{11}}\mathbf{J_{22}-\mathbf{J_{12}}\mathbf{J_{21}}} > 0$. The biological interpretation of this criterion will be posed as a problem at the end of the chapter. For now, Table 7.2 will suffice.

Table 7.2: Interpreting eigenvalues of a Jacobian matrix.
Eigenvalues	Interpretation
All real parts $< 0$	Globally Stable Point (Point Attractor)
Some real parts $< 0$	Saddle (Attractor-Repellor)
No real parts $< 0$	Globally Unstable Point (Point Repellor)
Real parts $= 0$	Neutral
Imaginary parts absent	No oscillations
Imaginary parts present ($\pm\omega i$)	Oscillations with period $2\pi/\omega$

Eigenanalysis of the Jacobian matrix in R is easy once we evaluated the Jacobian matrix at its equilibrium (above). We simply perform eigenanalysis on the matrix (from previous boxes: $\alpha_{11}=\alpha_{22}=0.01,\,\alpha_{12}=\alpha_{21}=0.001,\, r=1$).

eigStable <- eigen(J1); eigStable[["values"]]

## [1] -0.8181818 -1.0000000

The dominant eigenvalue is negative (the larger of the two: $\lambda_1$ = -0.818) indicating a globally stable equilibrium (Table 7.2. Both eigenvalues are real, not complex, indicating that there would be no oscillations (Table 7.2).

7.3.5 Three interesting equilbria

Here we examine the dynamical properties of three particularly interesting internal equilibria that are, respectively, stable, unstable, and neutral. In each case, our examples use $\alpha_{11}=\alpha_{22}=0.01$ and $r_1=r_2=1$. What is most important, however, is not the particular eigenvalues, but rather their sign, and how they vary with $\alpha_{12}$ and $\alpha_{21}$, and the resulting stability properties and trajectories.

Given our stability criteria above, let us next examine the dominant eigenvalue of the Jacobian for each equilibrium but which values of $\alpha_{ij},\,\alpha_{ji}$ should we choose? We can describe our invasion criterion for species $i$ as \[\begin{equation} \tag{5.7} \beta_{ij}=\alpha_{ij}/\alpha_{jj} \end{equation}\] where, if $\beta_{ij}<1$, species $i$ can invade. This ratio is the relative strength of inter- vs. intraspecific competitive effect. It turns out to be useful to calculate $\lambda_1$ (``perturbation growth rate’’) for combinations of $\beta_{ij},\,\beta_{ji}$.

Stable equilibrium – $\beta_{ij},\, \beta_{ji} <1$: These criteria correspond to $\alpha_{12}<\alpha_{22}\, , \, \alpha_{21}<\alpha_{11}$. As the relative strength of interspecific effects increases toward 1.0, $\lambda_1$ approaches zero, at which point the system would no longer have a single global point attractor.

When $\beta_{ij},\, \beta_{ji} <1$, then both species can invade each other. We find that all of the eigenvalues of the Jacobian are negative and real (Fig. 7.5, demonstrating that these populations will reach a stable equilibrium (Table 7.2. When we plot these eigenvalues for these combinations of $\beta$, we see that the dominant eigenvalue increases from negative values toward zero as either $\beta_{12}$ or $\beta_{21}$ approaches 1 (Fig. 7.5).

Stable equilibria: as the relative strength of interspecific competition increases (B(ij) increases toward 1), instability increases (lambda(1) increases toward 0). Solid dots represent initial abundances.

Figure 7.5: Stable equilibria: as the relative strength of interspecific competition increases (B(ij) increases toward 1), instability increases (lambda(1) increases toward 0). Solid dots represent initial abundances.

Unstable equilibria – $\beta_{ij}, \, \beta_{ji} > 1$

These criteria correspond to $\alpha_{12}>\alpha_{22} ,\, \alpha_{21}>\alpha_{11}$ (Fig. 7.6). As we saw above, the Lotka-Volterra competition model has not only stable equilibria, but also unstable equilibria, when both populations are greater than zero. Although an unstable equilibrium cannot persist, $\beta_{ij}, \, \beta_{ji} > 1$ creates interesting and probably important dynamics(Hastings 2004). One of the results is referred to as founder control, where either species can colonize a patch, and whichever species gets there first (i.e. the founder) can resist any invader (Bolker, Pacala, and Neuhauser 2003).

Another interesting phenomenon is the saddle itself; this unstable equilibrium is an attractor-repeller, that is, it attracts from some directions and repels from others (Fig. 7.6). This implies that the final outcome of dynamics may be difficult to predict from initial trajectories.

Unstable equilibria: as the relative strength of interspecific competition increases (B(ij) > 1), instability increases (lambda(1) > 0). The unstable equilibrium may attract trajectories from some initial states, but repel from others (solid dots represent initial abundances).

Figure 7.6: Unstable equilibria: as the relative strength of interspecific competition increases (B(ij) > 1), instability increases (lambda(1) > 0). The unstable equilibrium may attract trajectories from some initial states, but repel from others (solid dots represent initial abundances).

Recall the geometric interpretation of this unstable equilibrium — a saddle. The trajectory of a ball rolling across a saddle can depend to a very large degree on where the ball starts. Place it on the crown of the saddle, and it will tend to roll in a very deterministic fashion directly toward the unstable equilibrium, even if it eventually rolls off the side.

Eigenanalysis in R of the Jacobian where $\beta_{ij}, \, \beta_{ji} > 1$: Here we create values for $\mathbf{\alpha}$ that create an unstable equilbrium.

a11 <- a22 <- 0.01
a12 <- a21 <- 0.011
N1 <- eval(N1Star); N2 <- eval(N2Star)
eigen( eval(J) )[["values"]]

## [1]  0.04761905 -1.00000000

The dominant eigenvalue is now positive, while the other is negative, indicating a saddle (Table 7.2.

Neutral equilibria — $\beta_{ij} = \beta_{ji} = 1$: What happens when the inter- and intraspecific effects of each species are equal? This puts the populations on a knife’s edge, between an unstable saddle and a stable attractor. Let’s think first about a geometric interpretation, where we shift between a bowl, representing a stable attractor, and a saddle, representing what we call a neutral saddle.

Imagine that we begin with a stable attractor, represented by a bowl, where $\alpha_{ij} < \alpha_{ii}$. We drop a ball in a bowl, and the bowl rolls to the bottom — the global attractor. As we increase the interspecific competition coefficients, $\alpha_{ij} \to \alpha_{ii}$, we are pressing down on just two points on opposite sides of the bowl. Our hands push down on two opposite sides, until the bowl is flat in one direction, but has two remaining sides that slope downward. Perhaps you think this looks like a taco shell? The same shape is easily replicated by just picking up a piece of paper by opposite edges, letting it sag in the middle. This is the neutral saddle. What would eigenanalysis tell us? Let’s find out.

We could just charge ahead in R, and I encourage you to do so, repeating the steps above. You would find that doesn’t work because when $\beta_{ij} = \beta_{ji} = 1$, our equilibria are undefined (numerator and denominator are zero in (7.8), (7.9). Hmmm. Perhaps we can simplify things by taking the limit of the equilibrium, as $\alpha_{ij} \to \alpha_{jj}$. Let $\alpha_{12}=a$ and $\alpha_{22}=a+h$, and let $\alpha_{21}=b$ and $\alpha_{11}=b+h$. Then we want the limit of the equilibrium as $h$ goes to zero. \[\begin{align} \tag{7.16} \lim_{h \to 0}\frac{\left(a+h\right)-a}{\left(a+h\right)\left(b+h\right) - ab} &=\frac{1}{a+b} \end{align}\] Thus, $N_1^* = 1/(\alpha_{11}+\alpha_{22})$, assuming $\alpha_{12} = \alpha_{22}$ and $\alpha_{21}=\alpha_{11}$. Therefore, the equilibrium population size is simply the inverse of the sum of these coefficients.

Eigenanalysis in R of the Jacobian where $\beta_{ij} = \beta_{ji} = 1$: Here we create values for $\mathbf{\alpha}$ that create a neutral equilbrium.

a11 <- a21 <- 0.01; a22 <- a12 <- 0.015

We determine $N^*$ using (7.16) because the usual expression, (7.9), fails because the denominator equals 0.

N1 <- N2 <- 1/(a11+a22)
eigen( eval(J) )[["values"]]

## [1] -1  0

The dominant eigenvalue is now zero, indicating a neutral equilibrium (Table 7.2. The neutral nature of this equilibrium results in more than one equilibrium. Let’s try a different one, also on the isocline.

N1 <- 1/(a11); N2<-0
eigen( eval(J) )[["values"]]

## [1] -1  0

Again $\lambda_1=0$ so this equilibrium is also neutral.

When we perform eigenanalysis, we find that the largest of the two eigenvalues is zero, while the other is negative. This reveals that we have neither a bowl nor an unstable saddle, but rather, a taco shell, with a level bottom — a neutral saddle.

$*Trajectories of N(1),N(2) for $eta_{12} = eta_{21} = 1$. The entire isocline is an attractor, a neutral saddle, and the final abundances depend on the initial abundances and the ratio of $lpha_{11}/lpha_{22}$. The circle represents our one derived equilibrium.*$

Figure 7.7: Trajectories of N(1),N(2) for $eta_{12} = eta_{21} = 1$. The entire isocline is an attractor, a neutral saddle, and the final abundances depend on the initial abundances and the ratio of $lpha_{11}/lpha_{22}$. The circle represents our one derived equilibrium.

For example, if the populations start at low abundances, both populations will tend to increase at constant rates until they run into the isocline. Thus, both populations can increase when rare, but the relative abundances will never change, regardless of initial abundances.

Recall the Lotka-Volterra isoclines, and what we originally stated about them. We stated that the equilibrium will be the point where the isoclines cross. When all $\beta_{ij} = \beta_{ji} = 1$, the isoclines completely overlap, so we have an infinite number of equilibria—all the points along the line \[\begin{equation} N_2= \frac{1}{\alpha_{22}} - \frac{\alpha_{11}}{\alpha_{22}}N_1 \end{equation}\] and the initial abundances determine the trajectory and the equilibrium (Fig. 7.7).

7.4 Return Time and the Effect of $r$

Above, we referred to $\lambda_1$ as the perturbation growth rate. More commonly, people refer to another quantity known as characteristic return time. Return time is commonly calculated as the negative inverse of the largest real part of the eigenvalues, \[\begin{equation} \tag{7.17} RT = -\frac{1}{\lambda_1}. \end{equation}\] It is the time required to return a fraction of the distance⁴⁹ back toward an equilibrium. Negative return times ($\lambda_1>0$) refer to “backward time,” or time into the past when this population would have been this far away (Fig. 7.8.

If we envision the populations sitting at an equilibrium, we can then envision a small perturbation that shifts them away from that point in state space. Let’s call this displacement $x_0$. The rate of change of in $x$ is approximately the exponential rate, \[\begin{equation} \tag{6.5} \frac{d x}{d t} \approx c\lambda_1 t. \end{equation}\] where $c$ is a constant, so the distance traveled, $x$, is given by (7.15). Therefore, a negative $\lambda_1$ indicates an exponential decline in the disturbance, back toward the equilibrium (Fig. 7.8. The units of return time are the same as for $r$. Recall that all of this depends on the linearization of the curved surface around an equilibrium; it therefore applies exactly to only an infinitesimally small region around the equilibrium. It also usually provides the average, regardless of whether the perturbation is a population decline or a population increase.

$*For $eta_{ij} < 1$, return time is positive because some time will lapse before the system returns toward to its equilibrium. For $eta_{ij}>1$, return time is negative, because the system was closer to the (unstable) equilibrium in the past.*$

Figure 7.8: For $eta_{ij} < 1$, return time is positive because some time will lapse before the system returns toward to its equilibrium. For $eta_{ij}>1$, return time is negative, because the system was closer to the (unstable) equilibrium in the past.

Effect of $r$ on stability and return time

Consider the Jacobian matrix (7.14), and note that $-r_i$ appears in each Jacobian element. Therefore, the larger the $r$, the greater the magnitude of the Jacobian elements. This causes $\lambda_1$ to increase in magnitude, reflecting greater responsiveness to perturbation at the equilibrium (Fig. 7.9.

If we consider return time for continuous models where $\beta_{12},\,\beta_{21} < 1$, greater $r$ shortens return time, increasing stability (Fig. 7.9. For continuous models where $\beta_{12},\,\beta_{21} > 1$, greater $r$ increases perturbation growth rate, decreasing stability (Fig. 7.9. For discrete models, which we have not discussed in this context, recall that increasing $r_d$ of discrete logistic growth can destabilize the population because of the built-in lag. The same is true for discrete competition models — increasing $r_d$ too much destabilizes the interaction.

*The dominant eigenvalue of the Jacobian matrix varies with r as well as with $eta$ --- higher r causes greater responsiveness to perturbations around an internal equilibrium for r = 1 and r = 0.5.*

Figure 7.9: The dominant eigenvalue of the Jacobian matrix varies with r as well as with $eta$ — higher r causes greater responsiveness to perturbations around an internal equilibrium for r = 1 and r = 0.5.

7.5 Indirect competition for resources

Consumption is key to life. Energy consumption occurs as heterotrophs consume inorganic material and autotrophs use the sun or oxidation of inorganic molecules to acquire energy. All organisms require carbon and inorganic materials such as nitrogen and phosphorus as basic building blocks of cells. when one or more of these limit growth, we refer to these as limiting resources, and when differential growth limitation among species occurs as a result of this short supply, we refer to this as resource limitation and consequently resource competition.

In this chapter, we work through an example of resource competition among photoautotrophs for inorganic resources (Tilman 1982). As with nearly all the models in this book, We represent the growth of resources and consumers as growth rate = gains - losses. Here we start with a relatively simple version, with two plant species competing for one limiting resource such as nitrogen (Roughgarden 1998).

7.5.1 Linear and nonlinear responses

Let our resource $R$ be supplied at a constant rate, $S$, perhaps determined by nitrifying soil microbes at steady state. This rate declines as the amount of resource grows, or $S-aR$. This is likely to happen when a resource is washed out of the soil, or if a high amount of resource suppresses its production. \[\frac{dR}{dt} = S-aR\] In the absence of anything else removing the resource, the equilibrium amount would be $R^* = S/a$. However, our plant species will remove resource from the resource pool.

Let our competitors be measured in grams of biomass, where their growth rate depends on resource uptake and a constant loss rate, \[\frac{dP_1}{dt} = (u_1 R - d_1)P_1\] \[\frac{dP_2}{dt} = (u_2 R - d_2)P_2\] As we can see, these two species do not interact directly. Parameters $d_i$ are grams lost per gram of plant per unit time. The parameters $u_i$ are the rates of grams gained per gram of plant per unit resource per unit time. They do interact with resources. Here we graph mass-specific growth rates (analogous to per capita growth rates)as a funciton of resource concentrations.

u1 <- 1; d1 <- .3
u2 <- .5; d2 <- 0.1

{
  par(mar=(c(5, 4,.5,.5)))
  curve(u1*x - d1, ylab="dP/(Pdt)", xlab="Resource concentration")
  curve(u2*x - d2, lty=2, col=2, add=TRUE)
  abline(h=0, lty=3)
  legend("topleft", c("Sp. 1", "Sp. 2"), lty=1:2, col=1:2, bty='n')
  }

Figure 7.10: Mass-specific growth rates of two species across a range of resource concentrations.

From Fig. 7.10, we see that at high resource concentrations, species 1 grows faster than species 2. However, at low resource concentrations, species 2 grows faster than species 1. Resource concentrations change as plants take up resources, so when does the whole system reach an equilibrium?

Both species (Fig. 7.10) increase in biomass as long as resources are high enough to allow growth, whereas resources will decline as plant grow more and more and take up increasing amounts of resources. Eventually plant biomass is high enough to drive down resources so low that mass-specific growth rate is zero. This will happen at different resource concentrations for our two species. We see that species 2 has a positive growth rate when species 1 does not (Fig. 7.10).

If we try to solve for the equilibrium for $P_1$, we get \[\begin{align*} 0&=(u_1 R -d_1)P_1\\ 0&=(u_1 R - d_1)\\ R_{P1}^* &= d_1 / u_1 \end{align*}\] where we get an equilibrium resource concentration in the presence of only species 1, but not $P_1^*$. Similarly, we would find $R_{P2}^* = d_2 / u_2$. So, for the above species, the equlibirum resource concentrations are the $x$-intercepts, or

(Rstar <- c(R1=d1/u1, R2=d2/u2))

##  R1  R2 
## 0.3 0.2

Just as we saw in Fig. 7.10, these values predict that species 2 will grow at a lower resource levels than species 1 and is therefore predicted to exclude species 1. This type of resource competition for a single limiting resource was named the $R^*$ rule (“R-star”):

The species with the lowest $R^*$ for the shared limiting resource will exclude all others.

We refer to this type of resource competition as $R^*$ competition.

When one species excludes the other, what will the equilibrium value, $P^*$, be? For that, we need to refine our expression for $R$ to include uptake by plants: \[\frac{dR}{dt} = S-aR - w_1u_1P_1R - w_2u_2P_2R\] Parameters $w_1$ and $w_2$ are the amount of resource per gram of biomass, or the resource concentrations in the two plant species. Now that we have an explicit expression for resource dynamics, we can create a model to integrate the dynamics of these three components.

rcomp <- function(time, y, parameters){
  P1 <- y[1]; P2 <- y[2]; R <- y[3];
  with( as.list(parameters),{
    dP1 <- (u1*R - d1)*P1
    dP2 <- (u2*R - d2)*P2
    dR <- S -a*R - w1*u1*P1*R - w2*u2*P2*R
    return(list(c(dP1, dP2, dR)))
  })
}

Now we run the model.

t <- seq(0, 150, by=.5)
p <- list(u1=1, u2=0.5, d1=0.3, d2=0.1, w1=.01, w2=.01, S=1, a=.1)
Peq <- with(as.list(p),{
  c( P1=(S*u1 - a*d1)/(w1*d1*u1), P2=(S*u2 - a*d2)/(w2*d2*u2))
})
Peq

##       P1       P2 
## 323.3333 980.0000

y.initial <- c(P1=200, P2=50, R=1)
rout <- ode(y=y.initial, time=t, fun=rcomp, parms=p)
rdf <- data.frame( rout)
r.long <- pivot_longer(rdf, -time, names_to = "State", values_to = "Grams")
ggplot(data=r.long, aes(time, Grams, colour=State, linetype=State)) + 
  geom_line() + facet_wrap(~State, scales="free")

Now if we like, we can find $P_2^*$ when $R^*=d_2/u_2$, and $P_1 =0$. \[\begin{align*} \frac{dR}{dt} = 0 &= S-a(d_2/u_2) - w_2u_2P_2(d_2/u_2)\\ w_2d_2P_2 &= S-ad_2/u_2\\ P^*_2 &= \frac{S u_2 - a d_2}{u_2 w_2 d_2} \end{align*}\]

It is clear from this expression that increasing the death rate ($d$) will drive down $P^*_2$, and so will $w$ because it is less efficient—as the nutrient content of the plant increases, the amount of biomass it can make declines.

We also see that increasing the supply rate, $S$, and decreasing the background resource loss rate, $a$, will drive up biomass. In addition, as long as $S > w_2 d_2$ increasing resource-dependent gain rate, $u_2$ will also drive up equilibrium biomass.

# let u_2 be 'x'
# I use bquote() for math-friendly labels, but you don't have to.
with( as.list(p), {
curve( (S *x - a*d2)/(x*w2*d2),  0, 3,
       ylab=bquote((S *u[2] - a*d[2])/(u[2]*w[2]*d[2])), xlab=bquote(u[2]) )
})

Figure 7.11: As uptake rate ($u$) increases, euilibrium biomass will increase as well.

In the above model, per capita growth rates were linear functions of resource concentration. In Tilman’s presentation of resource competition, the consumers had nonlinear resource-dependent growth rates, \[\frac{dP_i}{Pdt}= r_i\frac{R}{k_i + R} - d_i\] You might recognize this as related to Michaelis-Menton enzyme kinetics. It was used in bacteriology by Monod to model culture growth in chemostats. It looks like Fig. 7.12.

r <-c(1, .5); k=c(5, .1); d = 0.3
{curve(r[1]*x/(k[1]+x) - d, 0, 10, ylab=bquote(dP/(Pdt)), xlab="R")
curve(r[2]*x/(k[2] + x) - d, lty=2, add=TRUE)
abline(h=0, lty=3); text(7.5, 0, "gain = loss", adj=c(.5, -.3))
}

Figure 7.12: The Monod function for per capita resource-dependent growth rates of two different populations.

As we saw earlier with linear mass-specific growth rates, one species will have a lower $R^*$ than the other, and this model predicts that it will outcompete the other.

7.5.2 Relative nonlinearity

The $R^*$ model does not permit coexistence in part because of how the resource replenishes. Our above model of the resource $R$ permits only a declining rate of change (rate of renewal declines with increasing R). In contrast, if the resource is biotic, such as a rapidly growing prey population, the renewal may accelerate with increasing population size.

*Different types of resources can renew in different ways. An inorganic soil nutrient might renew the way we modeled it in R* competition, whereas a prey population might renew via exponential growth.*

Figure 7.13: Different types of resources can renew in different ways. An inorganic soil nutrient might renew the way we modeled it in R competition, whereas a prey population might renew via exponential growth.*

Under these circumstances, it is possible for our model to predict coexistence, if the two competitors have sufficiently different curvature in their growth responses (Armstrong and McGehee 1980). This gives rise to the name “relative nonlinearity”.

Here is a sample model: \[\begin{align*} \frac{d P_1}{dt} &= m_1\left(-1 + \alpha_1 \frac{R}{k_1 + R} \right)P_1 \\ \frac{d P_2}{dt} &= m_2\left(-1 + \alpha_2 \frac{R}{k_2 + R} \right)P_2 \\ \frac{dR}{dt} &= rR(1-\alpha_3 R) - \alpha_1 P_1\frac{R}{k_1 + R} - \alpha_2 P_2\frac{R}{k_2 + R} \end{align*}\]

*The type of fixed oscillations possible given a self-reproducing resource and consumers with whose nonlinear growth rates differ in their curvature. The parameters are the same as in the previous figure.*

Figure 7.14: The type of fixed oscillations possible given a self-reproducing resource and consumers with whose nonlinear growth rates differ in their curvature. The parameters are the same as in the previous figure.

7.5.3 Competition for two resources: the resource ratio model

The $R^*$ model predicts no coexistence, and relative nonlinearity works with only a biotic resource. Tilman (1982) proposed the resource ratio model, that uses two resources with two or more competitors. In this model, what determines competitive outcomes are the ratios of the

supply rates of the two resources, and the
uptake rates of each species.

Coefficients reflect resource $i$ and species $j$.

\[ \frac{1}{R_1}\frac{dR_1}{dt} = a_1\left(S_1-R_1\right) - \left( dN_1/dt + m_1 \right) / e_{11} - \left( dN_2/dt + m_2 \right) /e_{12} \]

\[ \frac{1}{R_2}\frac{dR_2}{dt} = a_2\left(S_2-R_2\right) - \left( dN_1/dt + m_1 \right) / e_{21} - \left( dN_2/dt - m_2 \right) /e_{22} \]

\[ \frac{1}{N_1}\frac{ dN_1 }{dt} = r_1 \,\mathrm{MIN}\left(c_{11}\frac{R_1}{k_{11} + R_1} \, , \, c_{21}\frac{R_2}{k_{21} + R_1}\right) - m_1\]

\[ \frac{1}{N_2}\frac{ dN_2 }{dt} = r_2 \,\mathrm{MIN}\left(c_{12}\frac{R_1}{k_{12} + R_1} \, , \, c_{22}\frac{R_2}{k_{22} + R_1}\right) - m_2\]

This model might be used to represent algae competing for nitrogen and phosphorus. In our description below, we’ll use the example of eastern red cedar (Juniperus virginiana) and red maple competing for nitrogen and light.

Here we present one of the cases proposed by Tilman, in which resources are not substitutable. This means that getting more of one doesn’t help compensate for a shortage of the other. This is reflected in the zero net growth isoclines (ZNGI) in Fig. ??. Each ZNGI is perpendicular to the relevant resource axis. For instance, the ZNGI for light is flat; it doesn’t matter how much nitrogen maple gets, its minimum light requirement remains the same.

*Two resources, and the zero net growth isoclines for two competing species. These show that maple has a higher minimum requirement for nitrogen than for light, and the reverse for juniper. The arrows show how each species consumes the two resources simultaneously---they are the 'consumption vectors'. Their direction shows how maple and juniper drive down light and nitrogen levels. The different slopes show us that us that each species consumes the two resources at different rates. Maple consumes twice as much light as nitrogen, and reverse for juniper. The units are arbitrary.*

Figure 7.15: Two resources, and the zero net growth isoclines for two competing species. These show that maple has a higher minimum requirement for nitrogen than for light, and the reverse for juniper. The arrows show how each species consumes the two resources simultaneously—they are the ‘consumption vectors’. Their direction shows how maple and juniper drive down light and nitrogen levels. The different slopes show us that us that each species consumes the two resources at different rates. Maple consumes twice as much light as nitrogen, and reverse for juniper. The units are arbitrary.

If we put these species on the same graph, we get Fig. 7.16. Here we see what happens to light and N levels when both species are driving down resource levels at the same time: their consumption vectors sum so that the slope of the combined vector (grey) is intermediate between the two different species consumption vectors (black or dashed).

Fig. 7.16 helps illustrate three different environments: high light, low nitrogen (B), intermediate levels of both (A), and low light, high nitrogen (C). When both resources are supplied at intermediate rates (A), the juniper and maple drive down light and N until these resources limit both species simultaneously and the coexist.

If light is supplied at a much higher rate (B), the high light levels favor juniper because juniper can do very well if it gets lots of light. This illustrates a key insight from resource competition theory. Juniper succeeds in a high light envornment precisely it is a good competitor for resources other than light. In this example, juniper is not a good competitor for light. Light is supplied at a high rate. All species have enough light. The species that succeeds is the one that competes well for what is in shorter supply. To repeat that idea: the species that succeeds in a resource rich environment is not a good competitor for that resource. Instead, it is released from limitation by that resource. It may be a good competitor for some resource that is in short supply.

*Two resources, and the zero net growth isoclines for two competing species. This figure merely combines the two previous figures. As above, each species consumes more of the resource for which they have a higher requirement. A-C illustrate different resource supply rates. A, Light and nitrogen are supplied at equal rates. B, Light is supplied at a higher rate than nitrogen. C, Nitrogen is supplied at a higher rate than light.*

Figure 7.16: Two resources, and the zero net growth isoclines for two competing species. This figure merely combines the two previous figures. As above, each species consumes more of the resource for which they have a higher requirement. A-C illustrate different resource supply rates. A, Light and nitrogen are supplied at equal rates. B, Light is supplied at a higher rate than nitrogen. C, Nitrogen is supplied at a higher rate than light.

In contrast to the above scenario, species will not coexist if they consume more of the resource that is less limiting and less of the resource that is more limiting (Fig. 7.17). $\frac{R_{2,i}^*}{R_{1,i}^*}$

*What happens with greedy species. When species consume more of the resource for which they have lower reuirements, theory predicts competitive exclusion, where the outcome is determined by initial abundances.*

Figure 7.17: What happens with greedy species. When species consume more of the resource for which they have lower reuirements, theory predicts competitive exclusion, where the outcome is determined by initial abundances.

resratio <- function(t, y, parameters){
  R1 <- y[1]; R2 <- y[2]; N1 <- y[3] ; N2 <- y[4] ;
  with(as.list(parameters), {
    N1.dot <- r1 * N1 * ( min( c( c11*R1/(k11 + R1), c21*R2/(k21 + R2) ) ) - m1 )
    N2.dot <- r2 * N2 * ( min( c( c12*R1/(k12 + R1), c22*R2/(k22 + R2) ) ) - m2 )
    ## R.dot must follow N.dot because we use N.dot in R.dot
    R1.dot <- a1 * (S1 - R1) - (N1.dot + m1*N1)/e11 - (N2.dot + m2*N2)/e12 
    R2.dot <- a2 * (S2 - R2) - (N1.dot + m1*N1)/e21 - (N2.dot + m2*N2)/e22 
    return( list(c(R1.dot, R2.dot, N1.dot, N2.dot)) )
  } )
}

(#fig:pcgr.RR)Per capita growth rate.

Figure 7.18: Dynamics of competing species and the resources that they consume.

7.6 Summary

This chapter has provided several useful results.

We can represent species effects on each other in precisely the same way we represented their effects on themselves.
Considering only two species, species $i$ can invade species $j$ when the effect of species $j$ on species $i$ is less than its effect of species $j$ on itself.
Two species coexist stably when their effects on each other are smaller than their effects on themselves.
The dominant eigenvalue of the Jacobian matrix (perturbation growth rate), and its negative inverse, return time, are useful mathematical definitions of stability.
Perturbation growth rate decreases as $\beta_{ij},\, \beta_{ji}$ decrease, and are either both less than one or both greater than 1 ($\beta_{ij} = \alpha_{ij}/\alpha_{jj}$).
The magnitude of perturbation growth rate increases with $r$.

References

Armstrong, R A, and R McGehee. 1980. “Competitive exclusion.” The American Naturalist 115: 151–70.

Bolker, B M, S W Pacala, and C Neuhauser. 2003. “Spatial dynamics in model plant communities: {W}hat do we really know?” American Naturalist 162: 135–48.

Hastings, A. 2004. “Transients: the key to long-term ecological understanding?” Trends in Ecology and Evolution 19: 39–45.

Kingsland, S E. 1985. Modeling Nature. Chicago: University of Chicago Press.

Morin, P J. 1999. Community {E}cology. Malden, MA: Blackwell Science, Inc.

Roughgarden, J. 1998. Primer of {E}cological {T}heory. Upper Saddle River, NJ, USA: Prentice-Hall, Inc.

Tilman, D. 1982. Resource {C}ompetition and {C}ommunity {S}tructure. Monographs in Population Biology. Princeton, NJ: Princeton University Press.

Ecologists who study invasion of exotic species may use the word “invade” to mean the successful dispersal to, establishment and spread at a site. This incorporates a tremendous amount of species- or system-specific biology. Here we mean “invasion” in only a much more strict or narrow sense — to increase when rare.↩︎
The topography of mountain ranges can include saddles, which are precisely the context in which we use “saddle” here. Search the web for Saddleback Mountain, New York, USA, Lat/Long: 44$^\circ$ 8’ N; 73$^\circ$ 53’ W. See also pictures of horse saddles — same shape.↩︎
Recall that the time derivative, $d N / d t$, can be symbolized with $\dot{N}$ (“n-dot”).↩︎
PDEs are typically written using a fancy symbol, delta, as in $\partial F/\partial N$. It helps alert us to the fact that we are differentiating one variable with respect to another state variables. When taking a derivative with respect to $X$, we treat all other variables as constants.↩︎
This criterion is different than for demographic, discrete time projection matrices.↩︎
The magnitude or absolute value of a complex number $|a+bi|$ = $\sqrt{a^2 + b^2}$.↩︎
This fraction is about 63% or $1/e$; thus the hypothetical initial perturbation $x_0$ shrinks to $0.37x_0$.↩︎

Parameter	Description
\(r_i\)	Instantaneous rate of increase; intrinsic rate of growth; individuals produced per individual per unit time.
\(\alpha_{ii}\)	Intraspecific density dependence; intraspecific competition coefficient; the negative effect of an individual of species \(i\) on its own growth rate.
\(\alpha_{ij}\)	Interspecific density dependence; interspecific competition coefficient; the effect of interspecific competition; the negative effect that an individual of species \(j\) has on the growth rate of species \(i\).
\(K_i\)	\(1/\alpha_{ii}\); carrying capacity of species \(i\); the population size obtainable by species \(i\) in the absence of species \(j\).
\(\alpha^\prime_{ij}\)	\(\alpha_{ij}/\alpha_{ii}\); the relative importance of interspecific competition.
\(\beta_{ij}\)	\(\alpha_{ij}/\alpha_{jj}\); the invasion criterion for species \(i\); the relative importance of interspecific competition; the importance of the effect of species \(j\) on species \(i\) relative to the effect of species \(j\) on itself.

Primer of Ecology using R