Positive interactions

Commensalism (one species benefits, other unaffected)Mutualism (both benefit)

Benefit must be measurable at the population level:

Increase in one species brings about an increase in the per capita growth rate of the other

Graphic approach to Commensalism: the host

Combinations of densities of both species are shown on the graph. Arrows show regions that permit growth or force declines on Species 1.

The vertical line represents the combination of N1 and N2 that will result in N2 being constant-- the isocline for Species 2. Since Species 1 is not affected by Species 2, that line is vertical

N2

N1

K1

The Benefit

N2

N1

Species 2 benefits from species 1. We can envision this resulting from more resources, permitting more species 2 to be supported on a long-term basis. What is shown here is a facultative situation-- Sp. 1 is good for 2, but not essential. This is shown by K2: even with zero for N1, Species 2 can still grow to a carrying capacity.

However, life is better with Sp. 1 present, and the equilibrium for species 2 should increase as N1 increases, as shown by the upward-slope of Species 2’s isocline.

The isocline shows the combinations of N1 and N2 that will keepN2 constant. The region above shows density combinations thatforce declines in N2, and growth of N2 is possible in the regionbelow the isocline

K2

N2

N1

K2

Changes of both speciesNow that we’ve identified the density combinations that permitequilibrium, growth and decline, we can predict what will happen when the species interact by combining the two previous graphs:

K1

No matter where we start, we always end up at the intersection of the two isoclines--

Global stability

An obligatory commensalismIf some minimum density of the host is necessary for the commensal to grow at all, then its isocline might bend to pass through the axis of the host. The isoclines might be arranged like this.

N2

N1

K1

Note that it is possible for N2 to go to zero, if host density is low enough.

Minimum threshold of N1 needed for sp 2 to grow

Lotka-Volterra Competition Model

dN2

dt=r2N2 1−

N2

K2

−α21N1

K2

⎡

⎣ ⎢ ⎤

⎦ ⎥

dN1

dt=r1N1 1−

N1

K1

−α12N2

K1

⎡

⎣ ⎢ ⎤

⎦ ⎥

Equilibrium conditions, Species 1

y=a+bx

N2 =K1

α12

−1

α12

ˆ N 1

α12N2 =K1 − ˆ N 1

α12N2

K1

=1−ˆ N 1

K1

0=1−ˆ N 1K1

−α12N2

K1

0=r1ˆ N 1 1−

ˆ N 1K1

−α12N2

K1

⎡

⎣ ⎢ ⎤

⎦ ⎥

Isocline for Species 1:

N2

N1

K1

y=a+bx

N2 =K1

α12

−1

α12

ˆ N 1

N2 =K1

α12

−1

α12

ˆ N 1N2 =

K1

α12

−1

α12

ˆ N 1

Equilibrium Conditions, Species 2

ˆ N 2K2

=1−α21N1

K2

0=1−ˆ N 2K2

−α21N1

K2

ˆ N 2 =K2 −α21N

y=a+bx

0=r2ˆ N 2 1−

ˆ N 2K2

−α21N1

K2

⎡

⎣ ⎢ ⎤

⎦ ⎥

Isocline for Species 2:

N2

N1

K2

ˆ N 2 =K2 −α21Ny=a+bx

ˆ N 2 =K2 −α21N

K2

21

Outcomes