INTRODUCTION TO DYNAMIC MODELS
4.2 Dynamical Systems
Dynamical system models are the most commonly used type of dynamic model.
In a dynamical system model the forces of change are represented by differential equations. In this section we will focus on the graphical method for obtaining qualitative information about a dynamical system. The emphasis will be on questions of stability.
Variables: B = number of blue whales F = number of fin whales
gB= growth rate of blue whale population (per year) gF = growth rate of fin whale population (per year)
cB = effect of competition on blue whales (whales per year) cF = effect of competition on fin whales (whales per year) Assumptions: gB= 0.05B(1− B/150, 000)
gF = 0.08F (1− F/400, 000) cB = cF = αBF
B≥ 0, F ≥ 0
α is a positive real constant
Objective: Determine whether dynamic system can reach stable equilibrium starting from B = 5, 000, F = 70, 000
Figure 4.3: Results of step 1 for the whale problem.
Example 4.2. The blue whale and fin whale are two similar species that inhabit the same areas. Hence, they are thought to compete. The intrinsic growth rate of each species is estimated at 5% per year for the blue whale and 8% per year for the fin whale. The environmental carrying capacity (the maximum number of whales that the environment can support) is estimated at 150,000 blues and 400,000 fins. The extent to which the whales compete is unknown. In the last 100 years intense harvesting has reduced the whale population to around 5,000 blues and 70,000 fins. Will the blue whale become extinct?
We will use the five-step method. Notice that this problem is very similar to Example 4.1. Step 1 is to ask a question. We will use the number of blue and fin whales as state variables and make the simplest possible assumptions about growth and competition. The question we begin with is this: Can the two populations of whales grow to stable equilibrium starting from their current levels? The results of step 1 are summarized in Figure 4.3.
Step 2 is to select the modeling approach. We will model this problem as a dynamical system.
A dynamical system consists of n state variables (x1, . . . , xn) and a system of differential equations
dx1
dt = f1(x1, . . . , xn) ... ... dxn
dt = fn(x1, . . . , xn)
(4.3)
defined on the state space (x1, . . . , xn)∈ S, where S is a subset of Rn. The existence and uniqueness theorem of differential equations
states that if f1, . . . , fn have continuous first partial derivatives in a neighborhood of a point
x0= (x01, . . . , x0n),
then there exists a unique solution to this system of differential equa-tions through this initial condition. See any introductory text on differential equations for details (e.g., Hirsch, et al. (1974) p. 162).
Many other differential equation models can be reduced to the form (4.3). If the dynamics depend on time, we can introduce time as another state variable. If second derivatives are involved, we can include first derivatives as state variables, and so forth.
It is best to think of a solution to a dynamical system as a path through the state space. As long as differentiability assumptions are satisfied, there is a path through each point, and paths cannot cross except at an equilibrium. Let
x = (x1, . . . , xn) F (x) = (f1(x), . . . , fn(x)).
Then the dynamical system equation is dx
dt = F (x). (4.4)
For a path x(t), the derivative dx/dt represents the velocity vector.
Hence, for every solution curve x(t), we have that F (x(t)) is the velocity vector at each point. The vector field F (x) tells us in what direction and how fast we are moving through the state space. Usu-ally, a good idea of the qualitative behavior of a dynamical system in two variables can be obtained by drawing the vector field at selected points. The points where F (x) = 0 are the equilibria, and we will pay special attention to the vector field nearby these points.
Step 3 is to formulate the model. Let x1= B and x2= F , and write x′1= f1(x1, x2)
x′2= f2(x1, x2), where
f1(x1, x2) = 0.05x1
(
1− x1
150, 000 )
− αx1x2
f2(x1, x2) = 0.08x2
(
1− x2
400, 000 )
− αx1x2.
(4.5)
The state space is
S ={(x1, x2) : x1≥ 0, x2≥ 0}.
400,000
.08/α .05/α
150,000 x1
x2
Figure 4.4: Graph of fin whales x2 versus blue whales x1 showing the vector field (4.5) for the whale problem.
Step 4 is to solve the model. We want to sketch a graph of the vector field for this problem. Start out by sketching the level sets f1= 0 and f2= 0. The equilibria will be at the intersection of these two. Furthermore, the velocity vectors will be vertical along f1= 0 (x′1= 0) and horizontal along f2= 0 (x′2= 0). Draw the velocity vectors along these two curves. Then fill in some of the velocity vectors in between. It helps to remember that (as long as F (x) is continuous, which it usually is) both the length and direction of the vectors change continuously. In fact, for this kind of analysis, the length of the velocity vectors is not very important. See Figure 4.4 for the finished graph.
There are four equilibrium solutions, three at (0, 0)
(150, 000, 0) (0, 400, 000)
(4.6)
and another at a point whose coordinates depend on α. Our graph assumes that
400, 000 < (0.05/α).
In this case it is easy to see that the equilibrium in the interior is the only stable one. In fact, any solution through a point in the interior of the state space will eventually converge upon this equilibrium. In particular, the solution with initial conditions x1(0) = 5, 000 and x2(0) = 70, 000 tends to this equilibrium as t→ ∞.
Step 5 is to summarize the results of our model analysis in nonmathematical terms. Based on our analysis, in the absence of further harvesting, the whale populations will grow back to their natural levels, and the ecological system will remain in stable equilibrium.
Of course, our conclusions are based on some rather broad assumptions. For example, we have assumed that the effect of competition is relatively small. If it were larger (i.e., if (0.05/α) < 400, 000), then the two species could not coexist.
It does seem reasonable to make the assumption that α is small, since we know that the two species have coexisted for a long time before we began to harvest them. We have also made several simplifying assumptions about the growth process. The most critical of these is that for very small population levels, the population still tends to increase at the intrinsic growth rate. It is believed that some species have a minimum size (called the minimum viable population level) below which the growth rate is negative, ensuring the eventual extinction of the species. This assumption would, of course, change the behavior of our dynamical system. See Exercise 5 at the end of this chapter.
Finally, we address the questions of sensitivity analysis and robustness. First let us consider the sensitivity to the parameter α, for which we have very little information. For any value of α < 1.25× 10−7 there is a stable equilibrium x1> 0, x2> 0 at
x1= 150, 000(8, 000, 000α− 1) D
x2= 400, 000(1, 875, 000α− 1)
D ,
(4.7)
where
D = 15, 000, 000, 000, 000α2− 1,
which we found by Cramer’s Rule. For example, if α = 10−7, then x1=600, 000
17 ≈ 35, 294 x2=6, 500, 000
17 ≈ 382, 353.
(4.8)
The sensitivities at this point are
S(x1, α) =−21, 882, 352, 927
6, 000, 000, 000 ≈ −3.6 and
S(x2, α) = 27
221 ≈ 0.122.
The above calculations could be performed either by hand or by using a com-puter algebra system. Figure 4.5 illustrates the computation of the sensitivity S(x1, α) using the computer algebra system Maple.
> e1:=(5/100)*(1-x1/150000)-alpha*x2; -1 + 15000000000000 α2
, x1 = 150000 -1 + 8000000 α( ) -1 + 15000000000000 α2
-1 + 15000000000000 α2
- 4500000000000000000 -1 + 8000000 α( ) α
-1 + 15000000000000 α2
( )
Figure 4.5: Calculation of the sensitivity S(x1, α) for the whale problem using the computer algebra system Maple.
The blue whale population is most sensitive to α. If α = 10−8, then x1=276, 000, 000
1, 997 ≈ 138, 207 x2=785, 000, 000
1, 997 ≈ 393, 090.
Of course we will always have x1 < 150, 000 and x2 < 400, 000, as is apparent from the graph in Fig. 4.4. But the most important features of this equilibrium are not its coordinates, but rather the fact that an equilibrium exists on x1 >
0, x2> 0 and is stable. These conclusions remain valid over the entire range of α < 1.25× 10−7, which we believe to be plausible. Hence, we should say that our main conclusion is not at all sensitive to α. Similarly, our main conclusion is not at all sensitive to our data on intrinsic growth rate and carrying capacity, or even to the current whale populations.
Deeper questions of robustness revolve around the assumed form of the func-tions f1 and f2. We assumed that x′1/x1 and x′2/x2 are linear functions of x1 and x2, respectively. These lines represent the point where one species or the other stops growing. Suppose that we relax this linearity assumption. Let
x′1= x1g1(x1, x2) x′2= x2g2(x1, x2).
x1 x2
g1=0
g2=0
Figure 4.6: Graph of fin whales x2 versus blue whales x1 showing vector field for the generalized whale problem.
All of our analysis results would certainly remain true if g1 and g2 were not linear, so long as the vector field had the same general features. See Figure 4.6 for an illustration.