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Possible mini-projects
All projects involve finding equilibria and their stability. For the project you choose,
prepare a report summarizing your analysis and showing all parts of it. Provide as much
biological interpretation as you can of the equilibrium values, their stability, and the
dynamics as best you can determine them. In particular, for non-trivial, stable equilibria,
how does the equilibrium value depend on the parameters (i.e. what makes an
equilibrium value large or small)? Can you identify parameters that are stabilizing or
destabilizing as they change in value?
1. Chemostat / Resource models
The traditional chemostat model represents a microbial population that grows in relation
to a nutrient resource. In its original form, it can be written as two differential equations:
dN max RN

 DN
dt
KR
 RNq
dR
 D( S  R)  max
dt
KR
where the variables are
N(t)
R(t)
= microbial population density
= nutrient resource concentration
and the parameters are
max
= maximal growth rate
= half-saturation constant for growth
= dilution rate
= nutrient supply concentration
= nutrient quota per cell
K
D
S
q
All parameters are positive.
If we define the total nutrient in the system as the sum of the dissolved nutrient (R) and
the amount contained in microbial cells (Nq), then we can prove that as t → ∞, total
nutrient converges to a limit equal to the supply concentration, S. Therefore, for large
enough times, we can write
S  R  Nq
as a good approximation, which implies that
R  S  Nq
which we can substitute into the first differential equation above, to get
dN max ( S  Nq) N

 DN
dt
K  S  Nq
This enables us to write the chemostat model as a single differential equation.
Determine the equilibria and their stability for this one-dimensional model of the
chemostat equation.
In discrete time, we can derive resource-based models of population growth by the
following thought process, suggested by F.R. Adler. Start by writing a general recursion
equation
N (t  1)  N (t ) g ( R(t ))
where N is the population density and the function g(R(t)) is the per capita growth rate of
the population as a function of resource availability at time t, R(t). For simplicity, we can
assume that population growth is proportional to resource availability, so that
g ( R(t ))   R(t )
where  is a constant parameter. It is reasonable to assume that since organisms consume
the resource, resource availability must be a decreasing function of N(t). A simple
example is
R( N (t ))  e qN (t )
where q is a quota parameter governing how much the population decreases the resource
availability. Putting this together, a one-dimensional recursion equation is
N (t  1)   N (t )e qN (t )
To make biological sense, we should have q > 0 and  > 1.
Determine the equilibria and their stability for this one-dimensional, resource-based
model.
2. Fisheries models
In modeling fisheries it is often reasonable to assume that the rate of capturing fish is
related to the effort expended (E) and the population density of fish (N(t)), such that
rate of capture = -EN(t)
If we assume that a fish population grows according to the logistic model, then a model
of fishery dynamics can be written
dN
 N
 rN 1    EN
dt
 K
For this model the parameters r and K have the same names as in the logistic model,
intrinsic rate of increase and carrying capacity, and all three parameter are positive.
Find the equilibria and determine their stability for this continuous-time fisheries model.
Discrete-time models with a constant effort of fishing can be written many ways. One
that has a fairly long history is based on the Beverton-Holt model.
N (t  1) 
 N (t )
 EN (t )
1   N (t )
where all the parameters  and E are positive, and  > 1 for a biologically meaningful
model.
Find the equilibria and determine their stability for this discrete-time fisheries model.
3. Allee effects
Allee was an ecologist at the University of Chicago who pointed out a problem with the
logistic model for populations of sexually reproducing organisms. The logistic model
says that the rate of population growth is maximal when the population is very small
(infinitesimally near zero). But sexually reproducing organisms need to find mates for
reproduction and population growth to occur. So, below some threshold population, it
would be too hard to find a mate, and the population would decline instead of grow. So
Allee said that there should be a threshold population density (NT), such that if the initial
population was above the threshold, individuals would be able to find mates, and the
population would grow and reach its carrying capacity, K. For an initial population below
the threshold, finding mates would be so hard that the population would go extinct.
A continuous-time mathematical model consistent with the scenario that Allee proposed
can be written
dN
 aNT KN  a( K  NT ) N 2  aN 3
dt
where N(t) is the population density, and the parameters K and NT are the carrying
capacity and threshold population density. The parameter a has no simple biological
interpretation but governs how the rate of change is related to population density. For a
biologically reasonable model the parameters follow these constraints
a>0
and
K > NT > 0
Determine the equilibria and their stability for this continuous-time model of a population
with an Allee effect.
A discrete-time model consistent with the scenario that Allee proposed can be written
N (t  1)  1  aNT K  N (t )  a( K  NT )  N (t )   a  N (t ) 
2
3
The parameters have the same meanings as before, and follow the same constraints on
their values.
Determine the equilibria and their stability for this discrete-time model of a population
with an Allee effect.
4. Models for plant populations
Following theory developed primarily by J. Huisman, a continuous-time model for the
dynamics of plant biomass density (W, in g / m2) can be derived starting with the
assumption that the rate of photosynthesis at any height (s) within the vegetation depends
on the light intensity at that height (I(s)) according to
p( I ( s)) 
pmax I ( s)
h  I (s)
where pmax is the maximal rate of photosynthesis for saturating light intensity, and h is the
light intensity for which photosynthesis is half its maximal rate. We also assume that
plant biomass is uniformly distributed within the vegetation from height s = 0 (taken to
be the top of the canopy) to height s = smax (taken to be the soil surface), and that each
unit of plant biomass absorbs a certain proportion of the light coming from above so that
it shades the vegetation below. Taking this shading into account, we can integrate the
equation for p(I(s)) over the height of the vegetation to derive this differential equation:
dW pmax  h  I in

ln 
dt
k
 h  I out

  lW

where Iin is the light intensity at the top of the vegetation canopy and Iout is the light at the
bottom of the vegetation, at the soil surface, k is a parameter related to the proportion of
light absorbed by a unit of biomass and the vegetation height, and l is the loss rate of
plant biomass due to respiration, grazing by herbivores, and so on. Given the assumptions
of the model, light at the soil surface is
I out  I in e  kW
To complete the model, this last relationship has to be substituted into the differential
equation. Unfortunately, doing so leads to an equation that cannot be solved for
equilibrium algebraically. So consider the special case arising when vegetation is very
dense, so that Iout ≈ 0 (note that all parameters of the model are positive). Find the
equilibrium for this special case and determine its stability.
For 1000 extra points, can you prove that a nontrivial equilibrium exists for the general
case when Iout > 0? Is it stable? How does it compare in size to the nontrivial equilibrium
for the special case you analyzed? What can you say about a trivial equilibrium (note that
W = 0 implies Iout = Iin)?
A discrete-time model for population dynamics of an annual plant can be derived by
thinking about the seeds present during the winter (when only seeds exist). Let S(t)
represent the density of seeds present in winter, and assume that a fraction g of the seeds
germinate in spring (0 < g < 1), making gS(t) seedlings. From spring through summer
through autumn, the seedlings grow into plants which die off during the growing season
at an instantaneous rate that is proportional to the number of seedlings. Therefore, at the
end of a growing season that is T days long, the density of surviving plants is
gS (t )e dTgS (t ) , where d is the death rate of plants. In autumn, at the end of the growing
season, these survivors each produce F seeds. Therefore, in the next winter, the number
of seeds is the sum of those that did not germinate (which are assumed still to be viable),
and the new ones just produced:
S (t  1)  (1  g )S (t )  FgS (t )e dTgS (t )
Determine equilibria and stability for this discrete-time model of seed dynamics. All
parameters are positive and g < 1.