Transcript OBU Template

MATHEMATICAL MODELLING OF POPULATION
DYNAMICS IN THEORETICAL ECOLOGY
Department of
HARRY GREEN
Supervisors: DR. CRISTIANA SEBU, PROF. KHALED HAYATLEH, DR. TIM SHREEVE
The research project explores the use of ordinary
differential equations to model population change and the
effects of predation in multiple species ecosystems.
A system of 𝑛 coupled ODEs can be used to model a
system consisting of 𝑛 interacting species. These systems
are nonlinear and have the potential for chaos. The models
are essentially a development on the Lotka-Volterra
equations.
We will examine accuracy of the models by comparison
with real data using parameter fitting algorithms.
To model single isolated populations in the absence of
predation or competition we use a logistic growth curve
given by
𝑋
𝑋 = 𝑟𝑋 1 −
𝐾
(1)
where 𝐾 is the carrying capacity, an upper limit on the
species growth. In systems involving more species, logistic
growth will be used to simulate the growth of the species at
the bottom of the food chain (i.e., those that do not need to
feed on others to grow.
For predation terms, we use the ‘Holling Type II functional
response’ given by
𝑓 𝑥 =
𝑎𝑥
(2)
𝑏+𝑥
Where a is the maximum predation rate. 𝑓(𝑥) represents
the biomass consumed by each member of the predator
species y, which is converted into predator biomass with
efficiency c. Assuming predators die at some rate d, for a
two species food chain, we have the dynamical system
𝑋
𝑎𝑥
𝑋 = 𝑟𝑋 1 − 𝐾 - 𝑏+𝑥 𝑦
(3)
𝑎𝑥
𝑌=𝑐
𝑦 − 𝑑𝑦
𝑏+𝑥
Which can exhibit a range of behaviour including spiraling
and limit cycles. Bifurcation theory can reveal the existence
of a Hopf Bifurcation and show for which parameter values
the species can co-exist.
The stability of the fixed point at the point 𝐾, 0 determines
the long term survival of the predator.
Mechanical
Engineering and
Mathematical Sciences
The behaviour of a tritrophic food chain, when a
superpredator 𝑧 is introduced which feeds on the predator
𝑦, can be modelled by adding another dimension to (3).
The solutions become three dimensional, and we encounter
chaotic motion through a period doubling cascade.
The solutions demonstrate a sensitivity to initial conditions,
eliminating the possibility of any sort of long term prediction
of the ecosystem. As the system is also dissipative i.e., all
motion is bounded and all trajectories end in an attracting
set, we frequently find strange attractors.
𝑋
Modelling Human
population
growth with the
𝑋 = 𝑟𝑋
1−
(1)
𝐾
logistic curve (1). Data from UN estimates
A strange chaotic attractor plotted from a tritrophic food chain
model. The attractor demonstrates a sensitivity to initial
conditions, and a small change in parameter values changes
the shape unpredictably
Modelling global human population growth with the logistic
curve (1). Data from UN estimates and predictions. Human
growth is affected little by other species, and is a good
example of an isolated population.
Modelling Human population growth with the
Solutions of equation (3) for various initial conditions where the
curve (1).
Data
from
estimates
redlogistic
line is a boundary
on the
attracting
set,UN
calculated
from the
eigenvectors at (𝐾, 0) (K= 1 for this plot).
The models given here are only a small and simple subset
of the ODE based models used in theoretical ecology. Real
world species display a much broader range of behaviour
than simple Holling Type II predation.
Using non-autonomous systems of ODEs we can account
for seasonal variation in the system’s dynamics.
It is also frequently seen to combine the predator-prey
models with epidemiological models designed to model the
spreading of disease to investigate the effect of an
epidemic disease on a multiple-species ecosystem.