**Lecture 23 (Fr. 15-Mar-13) Predation (continued)**

**Return to Main Index page**
Go back to notes for Lecture 22, 13-Mar
Go forward to lecture 24, 25-Mar-13

Last time I introduced the topic of predation (types, importance and indirect effects) and explored:

1) Mesopredator effects on birds in California scrublands

2) Functional and numerical responses and the case history of Bay-breasted Warblers as predators of spruce budworm

Now we will turn to the classic Lotka-Volterra equations for jointly analyzing predator and prey dynamics.

Why analyze predators and prey jointly?

It seems reasonable to expect that abundant prey lead to increased predator numbers and that the converse is true for prey. We will concentrate on just a couple of models. The first will be unrealistically simple.**The Lotka-Volterra predator-prey equations/graphs** (continuous, differential equations)

We will make the following **assumptions**:

Exponential growth of the prey in the absence of predation(like Eqn 4.5 and Fig. 4.5). [dN/dt=rVpart of Eqn 23.1]

Linear functional responseby the predator (more prey mean more eaten) -- remember we had acurvedfunctional response in the Bay-breasted Warbler example (Fig. 22.1) [aVpart of Eqn 23.1. Look back atFig. 22.1to check the units and the axes for the functional response]

Numerical responseby the predator that is a constant times the functional response (tells us how prey consumption is converted into baby predators). Again, our previous numerical response example (Fig. 22.2) was a nonlinear curve. [b*(aV) part of Eqn 23.2]

Constant death rate of predators. [-dPpart of Eqn 23.2]{You may want to look at sections in the Gotelli text to get further info. on the concepts of functional and numerical responses by predators. We will return to, and modify, these responses in later analyses}.

We will give the victim population size (*V*) exponential growth minus a term due to predation loss.

Eqn 23.1

where *r* is the *V*ictim population growth rate, *aV* is the **functional response** of the predator and
*P* is the predator population size.

The equation for *P*redator population size (*P*) is:

Eqn 23.2

where *b* is the constant that turns the functional response into a **numerical response** (i.e., consumption into babies)
and *d* is the predator death rate. Note that if we take *b(aV)* as being a per capita birth rate of the predator,
we could rearrange Eqn 23.2 to get:

Eqn 23.3

which is simply the equation for exponential growth, with *P* instead of *N* as the population size variable.

**Solving for the isoclines:**

We plot the dynamics in a manner very similar to what we did for the competition equations. We look for the conditions where population size will not change (an equilibrium point) by setting d*Something*/d*t* to zero. In this case, we will set Eqns 23.1
(d*V*/d*t*) and 23.2 (d*P*/d*t*) to zero and solve for *V* or *P.* As for the competition equations, we use the resulting equations (23.4 and 23.5, below) to
draw *isoclines* (lines of equilibrium). Start with Eqn 23.1, the *V*ictim equation, to find the number of *P*redators required to cause no change in the number of
*V*ictims. We will call that equilibrium *P*redator number "*P*-hat", and write it as:

Because

Eqn 23.4

Do the same for Eqn 23.2.

Eqn 23.5

again, *d*, *a* and *b* are all constants, so we'll have a straight line perpendicular to the *V*ictim axis
(the *X*-axis). If the *V* are greater than the line *d/ab, *it will cause predators
to increase (more food converts into babies). *V* less than the line *d/ab* will cause predators to decline. We can put these
all together on a graph.

**Graphing the isoclines and using vector-sums to indicate joint trajectories:**

Here's the graph for the predator-prey isoclines. Note that I use the **vector-sum method** to
outline the dynamics of the four zones created by the isoclines.

The graph above hints at population cycles -- the prey shoot up, the predators then increase, the prey crash, the predators then crash and the prey then shoot up again.

Fig. 23.1.Predator-prey phase plot (prey,V, on theX-axis, predator,P, on theY-axis. Thepredator's isoclineis thegreenvertical line, labeledV-hat-- when prey are more abundant thand/ (ab) the predator increases (upward dashed arrows). When the prey are belowd/ (ab) the predator declines (downward dashed arrows). Theprey's isoclineis the horizontalredline, labeledP-hat-- with more thanr/apredators the prey decline (leftward dashed arrows). When predators are belowr/a,the prey increase (rightward dashed arrows). If we sum the vectors for predator and prey trajectories (angled, solidbluevectors) we see that they tend to orbit around the equilibrium isocline intersection.

**Phase plot of the joint trajectories**:

How would we go about calculating the joint trajectories of predator and prey? That requires solving the following differential equation:

Eqn 23.6

(Note that Eqn 23.6 is simply the left-hand sides of Eqns 23.1 and 23.2 on top of each other, with the d*t*
parts canceling). Solving Eqn 23.6 requires the technique called "separation of variables." (I used separation of variables to solve
d*N*/d*t* = *rN* for *N* in Eqns 4.6 to 4.12, and again to solve for *N* with the logistic equation).
Even with the solution via separation of variables we have an indefinite integral that cannot be solved explicitly --
because two solutions exist for every *X-value*, and two for every *Y-value* (as shown in the loop form of the trajectories
in Fig. 23.2, below).
Our competition trajectories didn't go round in orbits this way. We need to resort to numerical techniques, including what are
called interpolating functions. The mathematics for doing so is beyond the scope of this course.
(A computer program such as *Mathematica* can do most of the grunt work for us).
The resulting phase plot is shown in Fig. 23.2.

The behavior shown by Fig. 23.2 is sensitive to/dependent upon initial conditions. Previously, with our competition equation analyses, many different initial conditions would converge on a single equilibrium point (either stable coexistence or extinction of one or the other of the two species). Here, in contrast, every initial condition will have its own closed loop around which it cycles endlessly. A "movie" of the trajectories would show dots moving around each of the loops above in endless cycles. The outer loops will take longer to go all the way round and speeds will change in different parts of the loops (e.g., fast predator crashes on the left-hand edge of the outermost egg-shaped trajectory).

Fig. 23.2.Phase plot of predators against prey with isoclines like those of Fig. 23.1. Any initial condition (point in the phase plot representing the joint abundance of predator and prey) lies on a closed loop that will cycle in the same orbit (i.e., come back to its initial condition on each cycle). Each of the loops in the figure therefore results from a different initial condition. The outside loop started at {V= 20,P= 50}, the next at {120, 50}, then {220, 50}, {320, 50} and the innermost loop started at {420, 50}. The amplitude of the cycle (difference between maximum and minimum values) therefore depends upon the size of the initial deviation. Only an initial condition of {500, 50} would stay at the point of intersection of the two isoclines (because thatisthe equilibrium point that is the focus for all the orbits). Note that the trajectories are not circular because the prey grow more quickly than do the predators (maximum prey abundance is 5 times equilibrium, whereas maximum predator abundance is three times equilibrium). The cycle is: prey explosion, prey crash, predator crash (with a slower rise in the predators happening along the "width" of the egg-shaped orbit). Note that for these parameter values, and starting with just 20 prey as the initial condition, the predator crash occurs along a line that isvery close to prey extinction.

Parameters:r= 0.5,a= 0.01,b= 0.2,d= 0.1. [Check these parameters against the isocline solutions of Eqns 23.4 and 23.5].

I calculated the trajectories withMathematicaroutines, aided by theMathematica-based book: Gray, A., M. Mezzino, and M.A. Pinsky. 1997. Introduction to Ordinary Differential Equations withMathematica. Springer-Verlag, NY and itsODE.mpackage.

**Summary of Lotka-Volterra system of **Eqns 23.1 and 23.2:

• Simple (unrealistic) assumption ofReferences:exponential growth of preyin absence of predator (termrVin Eqn 23.1)• Simple (unrealistic) assumption of

linear functional response of predatorto prey density (termaVin Eqn 23.1)• Simple (unrealistic) assumption of

linear numerical responseof predator

(conversion of prey into baby predators -- termb(aV)Pin Eqn 23.2.

• Cyclical behaviorof predator and prey densities (reasonable)

butcycle amplitude determined entirely by initial conditions(unreasonable).

Bulmer, M.G. 1994. Theoretical Ecology. Sinauer Associates, Sunderland, MA.Gray, A., M. Mezzino, and M.A. Pinsky. 1997. Introduction to Ordinary Differential Equations with

Mathematica. Springer-Verlag, NY.Gurney, W.S.C., and R.M. Nisbet. 1998. Ecological Dynamics. Oxford Univ. Press, Oxford.

Return to top of page Go forward to lecture 24, 25-Mar-13