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**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

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**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 K1 N1

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The Benefit 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. N2 K2 N1 The isocline shows the combinations of N1 and N2 that will keep N2 constant. The region above shows density combinations that force declines in N2, and growth of N2 is possible in the region below the isocline

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**Changes of both species**

Now that we’ve identified the density combinations that permit equilibrium, growth and decline, we can predict what will happen when the species interact by combining the two previous graphs: No matter where we start, we always end up at the intersection of the two isoclines-- Global stability N2 K2 K1 N1

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**An obligatory commensalism**

If 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 Note that it is possible for N2 to go to zero, if host density is low enough. K1 N1 Minimum threshold of N1 needed for sp 2 to grow

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**Lotka-Volterra Competition Model**

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**Equilibrium conditions, Species 1**

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Isocline for Species 1: N2 K1 N1

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**Equilibrium Conditions, Species 2**

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Isocline for Species 2: K2 N2 K2 N1 a21

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Outcomes

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Two-species competition The Lotka-Volterra Model Working with differential equations to predict population dynamics.

Two-species competition The Lotka-Volterra Model Working with differential equations to predict population dynamics.

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