Project 1: Predator Prey Models
In this project you investigate and compare different predator prey models and their
dependence on parameters. We consider only one prey, but one or two predators. A
review of the basic models is attached. In the case of two predators you have to include
competition terms.
The two dimensional ODE models (one predator and one prey, continuous time, no
delay) can be conveniently solved using pplane6. For the 3d ODE systems (2 predators)
you have to use an ODE solver (suggestion: ode45) and the plotting tools of Matlab.
(a) Begin your investigation with the basic rescaled Lotka Volterra system, equation
(5) in the review. Create a phase portrait and some time plots for a fixed value of the
parameter. Convince yourself that the solutions are oscillating.
(b) Investigate the improved Lotka Volterra system, equation (8). This system shows
three types of behaviour:
(i) The predator dies out and the prey appoaches the carrier capacity.
(ii) The solution approaches a stable equilibrium with both predator and prey populations
nonzero.
(iii) The system shows a limit cycle oscillation, i.e. after a transient time the predator
and prey populations oscillate about an unstable equilibrium.
By experimenting with pplane6, find parameter values for which these types of behaviours
occur. In each case determine the stability properties of the equilibrium by computing
the eigenvalues of the Jacobean (this is easily done with pplane6).
(c) Simulate the discrete population model (13) for increasing values of r. Identify some
approximate values for which period doublings occur. Display the results in the form of
time plots and cobweb diagrams.
(d) Simulate the discrete predator prey model (12) for a > 1 and some selected (increasing) values of r. Display the results in the form of time plots ut and vt versus t. If
the motion appears chaotic, it is also instructive to plot the iterated points, after some
initial transient time, in a plane. Think about how the model can be extended to the case
of two competing predators.
(e) (Main part of project.) Extend the models of (a),(b) to the case of two predators
(P1 , P2 ), with competition terms proportional to −P1 P2 . The rescaled form of the extended Lotka Volterra system should read:
u0 = u − uv − uw
v 0 = bv(u − 1) − c12 vw
w0 = b2 w(u − 1) − c21 vw,
(1)
(2)
(3)
where u and v and w represent the prey and the first and second predators. For the
improved model you should obtain a system with the predation terms uv and uw replaced
by uvf (u, d) and uwf (u, d2 ) (see review).
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Investigate the effect of the second predator on the systems studied in (a),(b), i.e. fix
the parameters a, b, d as in (a),(b) and vary the remaining parameters. Examples of the
kind of questions which can be addressed are: When does the second predator loose or
win, and when is there a compromise? Does the presence of the second predator favor or
suppress oscillations? Can it happen that the competition causes both predators to die
out?
Display your results in the form of trajectories in the (u, v) and (u, w) planes and as
time plots u(t), v(t), w(t). In addition compute the equilibria with u∗ , v ∗ , w∗ all nonzero
and the eigenvalues of the associated Jacobean. Suggestion: For (1),(2),(3) the equations
for the equilibria reduce to a system of linear algebraic equations which is easily solved
with Matlab. For the improved three dimensional model, the equations v 0 = 0, w0 = 0
can be directly solved for v, w. Substituting these in u0 = 0 yields a single polynomial
equation for u. Polynomial equations are solved in Matlab via the roots command. The
eigenvalues of the Jacobean can be found using the eig command.
Optional: Study the DDE models (9), (10)-(11) using the solver dde23.
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Review
ODE Models
The basic predator prey model of Lotka and Volterra (see Murray, Sec. 3.2) is given by
N 0 = N (A − BP ), P 0 = P (CN − D),
(4)
where the prime denotes time derivative, N and P are the prey and predator populations,
and A, B, C, D > 0 are parameters. The rescaled version of this system reads,
u0 = u(1 − v), v 0 = av(u − 1),
(5)
with the rescaled prey and predator populations u and v and a single parameter a. System
(5) has a fixed point u∗ = v ∗ = 1, and all solutions with u(0) 6= 0 and v(0) 6= 0 oscillate
about this fixed point.
A more realistic predator prey ODE model (Murray, Sec. 3.3) is given by
N 0 = rN N (K − N ) − AN P f (N, D), P 0 = BN P f (N, D) − rP P.
(6)
The improvement consists of the carrier capacity K for the prey, and a more realistic
predation term N P f (N, D) (D is a parameter) replacing the simple product N P in (4).
We choose the following form for f ,
f (N, D) =
1
.
N +D
(7)
Note that N f shows a saturation, i.e. N f (N, D) → 1 when N → ∞. The rescaled version
of (6) reads (verify this)
u0 = u(1 − u) − uvf (u, d), v 0 = buvf (u, d) − av,
(8)
with new parameters a, b, d.
System (8) has an equilibrium P ∗ = (u∗ , v ∗ ):
u∗ =
ad
, v ∗ = (1 − u∗ )(u∗ + d),
b−d
provided b > d and a(d + 1) ≤ b. When b → a(d + 1), u∗ → 1 and v ∗ → 0, i.e. P ∗
approaches the carrier capacity of u. P ∗ is stable for d > 1. If d < 1, P ∗ is stable when
a > aH (b, d), where
aH (b, d) = (1 − d)(b − d)/d.
If d < 1 and a < aH (b, d), P ∗ is unstable, and the solutions evolve towards a stable limit
cycle surrounding P ∗ .
DDE Models
A further improvement is obtained by taking a delay into account. Since the predation
term of the predator equation contributes to the birth rate, we include the delay in this
term (for the prey predation acts instantaneously since it contributes to the death rate).
Thus N and P in the terms CP N in (4) and BP N f (N, D) in (6), are replaced by N (t−T )
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and P (t − T ). After rescaling this yields the following DDEs (delay versions of (5) and
(8)):
du(t)
dv(t)
= u(t)(1 − u(t)),
= au(t − T )v(t − T ) − av(t),
(9)
dt
dt
and
du(t)
= u(t)(1 − u(t)) − u(t)v(t)f (u(t), d)
dt
dv(t)
= bu(t − T )v(t − T )f (u(t − T ), d) − av(t).
dt
(10)
(11)
Discrete Models
In continuous models decay/growth of a population is modeled by a term of the form rP ,
with r > 0 for growth processes and r < 0 for decay processes. In discrete models, growth
or decay is distinguished by r > 1 and r < 1. The basic discrete predator prey model is
given by (Murray, Sec. 4.1)
Nt+1 = rNt exp(−APt ), Pt+1 = Nt (1 − exp(−APt )).
This model shows, however, unbounded growth for r > 1 and thus needs to be improved.
Unbounded growth can be avoided by inclusion of a carrier capacity for the prey:
Nt+1 = exp[R(1 − Nt /K) − APt ], Pt+1 = Nt (1 − exp(−APt )).
The rescaled version of this system reads
ut+1 = ut exp[r(1 − ut ) − avt ], vt+1 = ut (1 − exp(−avt )),
(12)
where r > 1. When vt = 0, (12) reduces to the one dimensional map
ut+1 = ut exp[r(1 − ut )].
(13)
This map has the fixed point u∗ = 1 which is stable for r < 2. At r = 2 there is a flip
bifurcation and for r > 2 we find a period doubling sequence leading to chaos, as for the
logistic map. The two dimensional map (12) has a fixed point P ∗ = (u∗ , v ∗ ), with u∗ and
v ∗ determined by
v ∗ = r(1 − u∗ )/a, (a + r)u∗ − r = au∗ exp[−r(1 − u∗ )].
The second equation is a transcendental equation for u∗ which cannot be solved analytically. The fixed point P ∗ exists for a > 1. When a < 1, P ∗ has disappeared and the
iterates vt always tend to zero for t → ∞. For a > 1, (12) shows cyclic as well as chaotic
behaviour.
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