**Lecture 25 (27-Mar-13) Harvest (a special form of predation)**

Return to Main Index page Go back to notes for Lecture 24, 25-Mar Go forward to lecture 26, 1-Apr

Harvest (for example, commercial harvest in a marine fishery, or hunting of game species) is a special form of predation. We will look at the effects of different kinds of harvest effort (constant, linear and sigmoid) on the dynamics of the harvested species.

Say
we have a harvested species (mule deer, elk, rainbow trout or a
commercially
important marine fish species). Assume it shows logistic population
growth.
That takes us back to Eqn. 5.1 and Fig. 6.1 --
look back at Fig. 6.1 to see what the plot of d*N*/dt as a function of *N* looks like. The d*N*/d*t*
(assuming *r* is positive) will lead to positive growth for 0 < *N* < *K*. Then we will take
away animals (*H*) by harvesting them in various ways -- constant harvest, linearly
increasing harvest and sigmoid harvest.

Eqn 25.1(= Eqn 5.1)

Your first reaction to the idea of plotting
logistic growth may still be to think of a sigmoid curve. Remember, however,
that the sigmoid curve is a plot of *N*(*t*) against *t*, the **solution** of the
differential equation Eqn 25.1. Here we want to plot d*N*/d*t* itself against population size (*N*).
When *N* is very small, d*N*/d*t* will be small. When *N* is large (nearing *K*) d*N*/d*t* will also be small.
d*N*/d*t* reaches its maximum at *K*/2. The result is the concave down
parabola shown in Fig. 25.1 (look back at
Lecture 6 notes).

Fig. 25.1.Population growth rate, dN/dt, of a harvested species (green curve, similar to the curve of Fig. 6.1) as a function of the population size of the harvested species (N,on theX-axis), based on the logistic equation (Eqn 5.1 and Eqn 25.1). The straight red horizontal line represents aconstant harvestnumber (H_{c}) that does not vary with density -- regardless of the size of the harvested population, the harvest will be 40 animals per unit time. The harvested species can maintain or increase its numbers only over the range of values where the green humped curve equals or exceeds the red line (in this case, forNbetween 200 and 800). We can find the value ofNthat yields maximum growth by taking the derivative of the right-hand side of Eqn 25.1 (I put that as a "question to ponder" in Lecture 5).

Parameter values:r= 0.25,K=1,000,H_{c}= 40.

**The effects of different kinds of harvest functions**

**First harvest scenario -- constant harvest (H _{c}). **

Fig. 25.2 is the plot of

Net growth is positive only for values of

Fig. 25.2. Net change (difference between dN/dtandH)in a harvested species under the fixed harvest,H_{c}, of Fig. 25.1. Note thatnetpopulation growth (logistic growth minus harvest) is negative forN> 800 andN< 200. This means that we have two equilibrium points (places where dN/dt = 0) just as we did for the "double-cross" case under the interspecific competition model. One of theses equilibria (atN= 800) is alocally stable equilibriumpoint (values ofN> 800 will decrease to 800, while values of 200 <N< 800 will increase toward 800. The other equilibrium point (atN= 200) isunstable-- values > 200 will move away toward 800, while values < 200 will move away toward 0. Maximal addition of animals under logistic growth alone was 60, but we are harvesting 40, so the net addition is 20. This curve has the same shape as that of Fig. 25.1 because all we have done is "pull" the green curve down by 40 at every point.

**Second harvest scenario -- linearly
increasing
harvest, H_{l}.**

Now let's change the harvest function, so that it responds to increasing population size of the harvested species. That is plotted as the dashed upward line in Fig. 25.3.

Fig. 25.3.Population growth rate of a harvested species (solid curve,Y-axis values) as a function of population size (N), based on the logistic equation (Eqn 5.1 and Eqn 25.1). The dashed line represents alinear harvestnumber (H). The harvested species can maintain or increase its numbers only over the range of values where the solid curve equals or exceeds the dashed line (in this case, for_{l}Nbetween 0 and approximately 680).

Parameter values:r= 0.25,K=1,000,H= 0.08N.

Now let's examine the **net** change in *N*.

By having the harvest relate to the population size (something most managers would clearly want to do) we have eliminated the unstable equilibrium (

Fig. 25.4. Net prey change (difference between dN/dtandHfor the linear harvest of Fig. 25.3. Note that_{l})netpopulation growth (logistic growth minus harvest) is negative forN> 680 but positive everywhere else. This means thatN*= 680 is agloballystable equilibriumpoint (values ofN> 680 will decrease to 680, while values ofN> 0 will increase toward 680. Note also that the point of maximum net growth is no longerK/2 -- instead it isN*/2. Maximal net addition is now 30.

**Questions to ponder**:

What would happen if we
increased the harvest
function so that it intersected the peak of the d*N*/d*t*
curve?

a) How would it affect size of harvest? How would
it affect the stability properties of the system?

b) What effect would if have on buffering against
environmental stochasticity?

**Third harvest scenario -- sigmoid
harvest function. **

Now for an interesting twist with a management
punchline.
Let's try a **sigmoid** harvest function -- harvest will start slow
for small population sizes, increase rapidly as the population
increases
and then level off at an asymptote. Why would a harvest function do this? At very small population sizes, hunters don't feel it is worth it even trying. As the harvestable population increases, hunter interest increases. At some point, though, everyone that has any interest in hunting is already doing so, so the harvest rate simply can't increase any further. That describes a sigmoid function: rapidly increasing at first, then increasing a slowing pace toward an asymptote.

The sigmoid harvest function is shown in Fig. 25.5.

Fig. 25.5.Logistic population growth rate of a harvested species (the "prey" -- green humped curve,Y-axis values) as a function of prey population size (N). The bluish sigmoidal line represents asigmoid harvestnumber (H_{S}). The prey can maintain or increase their numbers only over the range of values where the green curve equals or exceeds the bluish line (in this case, in two different zones of population size that we will examine below). Note thatH_{S}is a function ofN-- that is, we deal withH_{S}(N) rather thanN(t) orH(t). Parameter values:r= 0.25,K=1,000.

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This portion is not required knowledge, just for the interested.

Equation for sigmoid harvest (FYI, not required).

H(N) is the harvest at a population size ofN:Eqn 25.3

whereK_{H}is the asymptote for harvest,qis the "littler" growth rate of the harvest, and 1.1 isH(ø), the starting harvest for a small population of the harvested species. The corresponding differential equation for the rate of change of the harvest as a function of the change in the harvested species population size is:Eqn 25.4

We would solve this (to get Eqn 25.3) in a way similar to the way we went about solving d

N/dtforN(t) with either exponential or logistic growth equations, earlier in the course.

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Let's look at a plot of the **net** harvest function (d*N*/d*t*-*H*).

Fig. 25.6.Difference between dN/dtandHfor the sigmoid harvest of Fig. 25.5. Note thatnetpopulation growth (logistic prey growth minus sigmoid harvest) is positive in two different regions. Net growth is positive betweenN= 324 andN= 673, but negative above that point. This means thatN*= 673 is astable equilibriumpoint. BelowN' = 324 the population declines, soN' is anunstable equilibrium. BelowN= 141 the growth is again positive. This makesN'' = 141 asecond stable equilibrium. If the population ever dips belowN' it will settle on a new lower equilibrium, with lowerNand lower harvest. Note also that the point of maximum net growth is actually in the lower positive region (at a prey number,N, of approximately 40).

**Equivalence to the sigmoidal total
response seen
in Bay-breasted Warblers in Lecture 22. ** This is a different
example
of the possibility of multiple equilibria illustrated in Figures 22.1
to
22.4 of the first lecture on predation (Lecture 22). The
sigmoidal
predator response there also resulted in multiple equilibria
(alternating
between stable and unstable points). Look back at those notes and
make sure you understand the conceptual link.

**Note on population trajectory**:
remember, we
are not dealing with phase plots here, so the vector sum approach is
not
what we do. Instead, we look at the impact of the difference between
the
functions (d*N*/dt - *H*) on *N*. If the sum is
positive,
*N*
increases (you could put the rightward arrows right along the orange
curve
instead of along the *X*-axis). If the sum is negative, *N*
decreases
(you could put a leftward arrow along orange curve instead of
along
the *X*-axis).

**Why not manage for a linear rather than
a sigmoid
harvest function, since the linear function doesn't have the multiple
equilibria?**
The shape of the harvest function may not be something that the
managers
can really control. At the lower end, managers can't make people
hunt, fish or commercially harvest if they simply don't think it's
worth
it for little reward. At the upper end, economics, a limited number of
harvesters or some other factor may put a cap on the effort. The
sigmoid
harvest function may be more a fact of life than something one can
easily
manipulate. Of the three curves, it may be the most plausible -- and it
can have the scary, puzzling consequence of an "irreversible" decline
in
the stable population size.

**Points to ponder:**

1) Alternating stable and unstable equilibria: I have already mentioned an interesting pattern that emerges when multiple equilibrium points exist. With multiple equilibrium points (such as here where the line squiggles back and forth across the zero line, and each crossing point is an equilibrium of zero net change) the system will alternate between stable and unstable equilibrium points. If you think about which way the arrows point, you will realize that this pattern is inevitable, At each crossing, the arrows will flip directions. That inevitably creates an alternation between stable (arrows point in to zero crossing) and unstable (arrows point away from zero crossing). Although it is not mathematically similar, a somewhat analogous alternation of effects occurs in some trophic interactions (look back at the sigmoid functional response curve and the coyote, scrubland bird examples of Lecture 21 and forward to the examples in Lecture 29 on regulation and trophic interactions).2) Why might it be more likely that we would have

spruce budworm outbreaks (escape from regulationby predators) thanpermanent depression of the equilibrium population size (

establishment of regulationby predators) of ungulates

by large predators (hunters, wolves)?

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