**Lecture 20 (Wed. 6-Mar-13) ****Analysis of interspecific competition by means of paired continuous logistic growth equations.
The Lotka-Volterra competition equations.**

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When I discussed competitive exclusion I said that the most interesting case is when two competitors manage to
coexist. We will now look at models for **competitive coexistence** using systems of paired logistic equations.

To work with a simple model of competition between two species we will return to a modified form of the **continuous**
logistic equation (compare and contrast this with** Eqn 5.1 of Lecture 5**).

Eqn 20.1

where the subscript 1 refers to Species 1 and the 2
to Species 2. The g symbol (this is the Greek letter gamma) describes the nature of the competition
between the species -- we will get to that in more detail shortly. Notice that if we took out the g_{12}*N*_{2} term and removed the subscripts, we'd have the simple form of **continuous logistic equation **that we had in Eqn 5.1. Eqns 20.1 and 5.1 are
forms of logistic growth that **cannot** overshoot the carrying capacity because the density dependence [= response to crowding] is **instantaneous** -- a**s opposed to the discrete logistic** [Eqn 16.1] that we analyzed for the contest vs. scramble competition section. With the
discrete logistic, overshooting and chaotic dynamics were possible because of **time
lags**. Whereas Eqn 20.1 describes the dynamics of Species 1, a
very similar equation describes the dynamics of Species 2:

Eqn 20.2

Let's look back at the interaction term g_{ij}. What the interaction term tells us is how strong an effect Species *j* has
on growth of Species *i*, relative to the effect of *i* on itself.
For example, if g_{12} = 0.1, then every additional individual of Species 2 has one tenth as much
effect on Species 1 as would adding another individual of Species 1 --
that means that Species 2 doesn't have all that big a detrimental effect
on Species 1.

**Possible outcomes of the paired logistic equation model:**

It turns out that the interactions described by Eqns. 20.1 and 20.2 have three possible outcomes:

What we will see is that the outcome depends on ratios of the carrying capacities,1) Stable coexistence(both species persist, as in Fig. 20.1)

2) Unstable coexistence(one species will "win" but which wins depends upon initial conditions or the direction of even the slightest perturbation from an unstable equilibrium for coexistence)

3) Competitive exclusion

a) Species 1 always wins

b) Species 2 always wins

**Stability analysis -- where
does
the system have no tendency to change? **

Let's say we have the following values for the
parameters
*r* and *K*:

* r*_{1}
= *r*_{2} = 0.5, and *K*_{1} = *K*_{2}
= 1000.

Assume also that g_{12
}=g_{21} = 0.67. First, we'll set both equations to zero -- in other words, we
will ask "under what conditions will each of the species have no tendency to
change population size?" (we'll be doing a **stability analysis**).
Setting d*N*/d*t* in Eqns 20.1 and 20.2 to zero, we can
divide out the *r*_{1}*N*_{1} term (start with Eqn
20.1; zero divided by *r*_{1}*N*_{1} is
still zero) to yield:

Eqn 20.3

which simplifies to (the double arrows in the middle of each line mean that we are moving from one form of the equation to a rearranged or simplified version of the same equation):

Eqn 20.4

**From equations to graphs --
plotting
isoclines (meaning lines of equilibrium or zero change). ** Eqn 20.4 describes a straight line on a plot of *N*_{1} against *N*_{2},
as shown in Fig. 20.1. [That is, Eqn 20.4
has the general form *Y* = *b* + *mX*]. The line
describes the value of *N*_{2} for which *N*_{1} is
at equilibrium (has no tendency to change -- **but note that** along
most of the line Species 2 will not be at equilibrium, so the size of *N*_{1} can be changed by changes in
*N*_{2}). Let's look at the extremes.
If *N*_{1} is zero, then *N*_{2} will be at a *Y-intercept* value of
*K*_{1}/g_{12 } = 1,500. Along the other axis, *N*_{2}
will equal zero when *N*_{1} = 1,000 (look at the left-most version of Eqn 20.4 to convince yourself
of that -- only when *N*_{1} = *K*_{1} will that
denominator go to zero). So we will have a line from a *Y-intercept* of *N*_{2} = 1,500
(=*K*_{1}/g_{12})
to an *N*_{1} value of 1,000, as shown below. This line is
called the *N*_{1} ** isocline**.

1) What is happening at {

2) Why an equilibrium at {

Now do the same thing for Eqn 20.2 (i.e., set d*N*_{2}/d*t* to zero) to get:

Eqn 20.5

(You should do the algebra to show that this
equation
is correct, by setting Eqn 20.2 to zero and then solving for *N*_{2}).
This will be a line from *N*_{2} = 1,000 on the *Y-intercept* to *N*_{1} = 1,500 on the *X*-axis. This line is
called
the *N***_{2}
isocline**. In this case we
move from a "pure" single-species logistic (

{The reason I solved for

Now plot the two linear equations on a graph that shows the density of *N*_{1} against *N*_{2}.

Let's look at the plot of the isoclines on a population size graph.

Fig. 20.1Population plots for paired equations of continuous logistic population growth under interspecific competition. The red line depicts the isocline for Species 1 (dN_{1}/ dt= 0 from Eqn 20.1, as solved in Eqn 20.4), while the blue line depicts the isocline for Species 2 (dN_{2}/dt= 0 from Eqn 20.2 as solved in Eqn 20.5). The isoclines are given by Eqns 20.4 and 20.5. Note that the isoclines intersect at approximatelyN_{1}=N_{2}= 600. The thin black line connects the two carrying capacities and shows a totalN_{1}+N_{2}= 1000 (K_{1}=K_{2}= 1,000) --

I will refer to this line connecting the two carrying capacities as the "K-connector".

The fact that the isoclines crossabovetheK-connectorline means that we have a greater TOTAL number of animals at equilibrium (1,200) than we would have if just one or the other species were present (1,000).

Parameter valuesinserted into Eqns 20.1 and 20.2:r_{1}=r_{2}= 0.5;K_{1}=K_{2}= 1000; g_{ 12}= g_{21}= 0.67. The arrows represent a vector field analysis showing the size and directionality of two-dimensional population change.

**Conclusion** from Fig. 20.1 --
with values such that:

K_{1}/g_{12}>K_{2}andK_{2}/g_{21}>K_{1 }Eqns 20.6

** stable coexistence **is
possible
despite the interspecific competition

(i.e., despite the fact that the niches overlap). {We will add Eqns 20.6 to a
set of inequality "rules" in Eqns 20.1 that will delimit the major possible
outcomes of stable coexistence, unstable coexistence and competitive exclusion}.

**Total carrying capacity**. Note that the equilibrium point (intersection
of the red and blue isoclines) is **above** the black line that shows
a combined *N*_{1}+*N*_{2} of 1,000. That means
that with this kind of competition the total number of animals in the habitat
is larger than the total if only one of the two species were present. {With
curved isoclines, we could get a combined carrying capacity that was below
the black combo. line -- we'll see examples of that later}.

We'll begin by redoing Fig. 20.1 as an animated cartoon.

Below is an example of several starting points
for
Species1/Species 2 plots, and their **trajectories** over time
toward
the equilibrium for the case of **stable coexistence**. The yellow
dots,
for example, represent the path along which one of the dots in the
"movie"
moved towards the equilibrium point where the two isoclines intersect.

Fig. 20.2 Stable coexistence case. Trajectories through time for three different starting points of competition equations based on modified continuous logistic equations (Eqns 20.1 and 20.2). The three starting points were: yellow --N_{1 }= 800,N_{2 }= 20; green --N_{1 }= 1,300,N_{2 }= 500; blue --N_{1 }= 200,N_{2}= 700. Dots that are far apart represent very rapid movement; dots closer together represent slower rates of change (the time interval between dot "paintings" is constant, so fast rates make jumps, whereas slow change makes a smooth band). Note that the first green dot jumpsveryquickly to a value on the red isocline, then moves more and more slowly toward the equilibrium. Note also that the yellow trajectory takes a sharp corner. At first it moves quickly toward the red isocline (andN_{1}grows rapidly), then it follows theN_{1}isocline fairly closely toward the equilibrium point (so thatN_{1}isdecreasing).

Note the characteristic that the "visiting team" (N_{1}) has a higher intersection with theY-axis than the "home team" (N_{2}). The "visiting team" (N_{2}) is also further out theX-axis than the "home team's"K. The inequalities of Eqns 20.6 satisfy the conditions for stable coexistence.

Parameter values:r_{1 }=r_{2 }= 0.5;K_{1}=K_{2}= 1,000; g_{12 }= g_{21}=0.667.Go to "movie" of stable coexistence starting with low numbers of each species.

Go to "movie" of stable coexistence starting with 800

N_{1}and 50N_{2}(the yellow arc in Fig. 20.2).Here's a similar set of trajectories for the

unstable equilibriumcase.

Fig. 20.3.Unstable coexistence case. Trajectories through time for three different starting points of competition equations based on modified continuous logistic equations (Eqns 20.1 and 20.2). The three starting points were: yellow --N_{1 }= 500,N_{2 }= 1,200; green --N_{1 }= 1,300,N_{2 }= 400; blue green --N_{1 }= 200,N_{2}= 600. Dots that are far apart represent very rapid movement, those closer together represent slower rates of change. Note that the first yellow dot jumpsveryquickly to a value between the isoclines, then moves more and more slowly toward the equilibrium extinction ofN_{1}and persistence ofN_{2}atK_{2}. Note also that the green trajectory takes a corner. At first it moves quickly toward the red isocline (andN_{1 }decreases rapidly), then it follows theN_{1}isocline fairly closely toward extinction ofN_{2 }and persistence ofN_{1 }atK_{1}.

Note the characteristic that the "visiting team" (N_{1}) has alowerintersection with theY-axis than the "home team" (N_{2}). Likewise, the visitors (N_{2}) have a less-than-K_{1}isocline crosspoint on theX-axis.

Parameter values:r_{1 }=r_{2 }= 0.5;K_{1}=K_{2}= 1,000; g_{12 }= g_{21 }=1.5.Go to "movie" of unstable coexistence starting with 500

N_{1}and 1,200N_{2}.

Note that for the unstable case, if either species exceeds some threshold population size it will drive the other species
to (local) extinction. N.B. that if the parameters (*r*, *K*,
g) are exactly equal as in our simplest case then the species with higher
population size "wins". If the parameters are not exactly equal, then the threshold ratio may be different from 50:50.

Conclusion from Fig 20.3: if

* K*_{1
}/g_{12}
< *K*_{2} **and**
*K*_{2} /g_{21}
< *K*_{1 }
**Eqns 20.7**

then the coexistence is *UNSTABLE*
(the only parameters we have changed between Fig. 20.2 and 20.3 are the
two g_{ij
}values, BUT THAT CAUSES THE INEQUALITY SIGNS TO POINT
IN THE OTHER DIRECTION). Thus, the inequalities of Eqns
20.7 satisfy the conditions for **unstable coexistence**. {We will
add Eqns 20.7 to a set of inequality "rules" in Eqns 21.1 that will
delimit the major possible outcomes of stable coexistence, unstable coexistence
and competitive exclusion}.

What we have done in the material covered here is to find conditions under which different outcomes will occur. We have done so by means of a combination of equation-based analyses and graphical arguments.

Although we need to use the equations to solve the
stability analysis, it is easier (at least for me) to use the graphs and
some fairly simple rules-of-thumb to decide which of the three outcomes
we have for a particular combination of *r*, *K* and g values.

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