16.6: Single-resource phase space

Some aspects of competition for a resource are clarified by looking at the phase space, as introduced in Chapter 10. Combining Equations 16.3.1 and 16.3.2 gives the following as a starting point:

\[\frac{1}{N_1}\frac{dN_1}{dt}\,=\,m_1(R_{max}\,-\,R_1^{\ast})\,-\,u_1m_1N_1\,-\,u_2m_2N_2\]

\[\frac{1}{N_2}\frac{dN_2}{dt}\,=\,m_2(R_{max}\,-\,R_2^{\ast})\,-\,u_2m_2N_2\,-\,u_1m_1N_1\]

As before, \(m_i\) is the rate of growth of Species \(i\) for each level of resource above its minimum resource requirement \(R_i^{\ast}\), and \(u_i\) is the amount of resource tied up in each individual of Species \(i\). For reference, here is the assignment of parameters in terms of \(r_i\) and \(s_{i,j}\).

\[r_1\,=\,m_1(R_{max}\,−\,R_1^{\ast}),\,\,\,s_{1,1}\,=\,−u_1m_1,\,\,\,s_{1,2}\,=\,−u_2m_2\]

\[r_2\,=\,m_2(R_{max}\,−\,R_2^{\ast}),\,\,\,s_{2,1}\,=\,−u_1m_1,\,\,\,s_{2,2}\,=\,−u_2m_2\]

Where in the phase space will the growth rate be 0 for each species? For Species 1 it will be where

\(\frac{1}{N_1}\frac{dN_1}{dt}\,=\,0\,=\,r_1\,+\,s_{1,1}N_1\,+\,s_{1,2}N_2\)

Solving for \(N_2\) gives

\[N_2\,=\,−\frac{r_1}{s_{1,2}}\,-\frac{s_{1,1}}{s_{1,2}}N_1\,\,\,\,\,\leftarrow\,Species\,1\,isocline\]

Figure \(\PageIndex{1}\). Species 1 increases below its isocline, shaded with gray copies of the numeral 1.

Anywhere along that line, the population of Species 1 will not change, but on either side of the line it will (Figure \(\PageIndex{1}\)). Formulae for the four possible equilibria and their stability are in Table 10.1. The vertical intercept of the isocline, where \(N_1\,=\,0\), is \(-r_1/s_{1,2}\), and the horizontal intercept, where \(N_2\,=\,0\), is \(-r_1/s_{1,1}\). The slope is \(-s_{1,1}/s_{1,2})\,=\,(u_1m_1)/(u_2m_2)\).

Similarly, growth of Species 2 will be 0 where

\(\frac{1}{N_2}\frac{dN_2}{dt}\,=\,0\,=\,r_2\,+\,s_{2,2}N_2\,+\,s_{2,1}N_1\)

Figure \(\PageIndex{2}\). Species 2 increases below its isocline, shaded with gray copies of the numeral 2.

Solving for \(N_2\) gives

\[N_2\,=\,-\frac{r_2}{s_{2,2}}\,-\frac{s_{2,1}}{s_{2,2}}N_1\,\,\,\,\,\,\leftarrow\,Species\,2\,isocline\]

Again, anywhere along that line the population of Species 2 will not change, but on either side of the line it will (Figure \(\PageIndex{2}\)). The vertical intercept of that line, where \(N_1\,=\,0\), is \(-r_2/s_{2,2}\), the horizontal intercept, where \(N_2\,=\,0\), is -\(r_2/s_{2,1}\), and the slope is \(-s_{2,1}/s_{2,2}\,=\,(u_1m_1)/(u_2m_2)\).

Figure \(\PageIndex{3}\). Single-species parallel isoclines. Each species increases only below its respective isocline, shaded with gray with the species number, 1 or 2.

Notice this: In terms of the resource, the slope of the isocline for Species 2 is identical to the slope for Species 1—both are equal to (\(u_1m_1)/(u_2m_2\)). What does this mean? It means that the two isoclines are parallel. And that, in turn, means that the two species cannot permanently coexist.

The populations can fall into only one of the three regions of Figure \(\PageIndex{3}\). If they start in the upper region, they decrease until they enter the middle region. If they start in the lower region, they increase until they also enter the middle region. Once in the middle region, only Species 2 increases. That means the population of Species 1 is driven leftward, toward lower values of \(N_1\), while the population of Species 2 is driven upward, toward higher values of \(N_2\).

Figure \(\PageIndex{4}\). Flow across the phase space, as explained in Chapter 10, converging on a stable equilibrium where Species 2 excludes Species 1. \((r_1\,=\,0.75,\,r_2\,=\,0.52,\,s_{12}\,=\,−1.875,\,s_{21}\,=\,−0.533,\,s_{11}=s_{22}=−1)\).

These dynamics show up in the flow diagram of Figure \(\PageIndex{4}\). The origin (0,0) is an unstable equilibrium. In this single-resource system, any populations near extinction, but not completely extinct, increase until they hit the middle region. The horizontal axis has another unstable equilibrium, where Species 1 is at its carrying capacity and Species 2 is extinct \((−r_1/s_{1,1},\)0). Any populations near that unstable equilibrium soon arrive in the middle region. All populations not precisely on one of those two unstable equilibria converge on the red disc on the vertical axis, where Species 2 is at its carrying capacity, \(K_2\,=\,−r_2/s_{2,2}\), and Species 1 is extinct (0,\(−r_2/s_{2,2}\)). This equilibrium is called a “global attractor.”

Phase spaces thus provide another view of competitive exclusion, the theory of which applies at least to two species competing for a single resource at equilibrium.