Download PowerPoint slide show on ecological modelling concepts

Modelling concepts
• Modelling in discrete time (difference equations,
also known as updating equations)
• Modelling in continuous time (differential
equations)
• State variables – the quantities we wish to model
• Initial conditions – and their importance
• Biological processes - modelled mathematically
Gurney and Nisbet, Chapter 1
More concepts
• Deterministic models: if we know current
conditions of a system, we can predict its
future
• Stochastic models
• Balance equations: e.g. mussel population:
N t 1  N t  settlement  death
(See Excel file)
Types of solutions
• Analytic solutions
• Numerical solutions: obtained by repeated
application of an update rule; easy for
discrete time models – more difficult for
continuous time models.
• Qualitative solutions (rather than complete
solutions) sometimes useful
Equilibrium and Stability
Equilibrium
• Concept of equilibrium
• Notion of a “steady-state”
• Situation in which levels of state variables
remain in a state of no change through time
Equilibrium and Stability
Stability
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•
•
•
Stable and unstable processes
Attractors
Repellers
Equilibrium state:
– stable
– unstable
• Stability:
– global
– local
Simple dynamic patterns
Discrete time geometrical (exponential)
growth
• Geometric growth: Xt+1 = RXt
• Also called exponential growth (although,
strictly, this is a continuous time concept):
Alternative forms of equation for
discrete exponential growth
•
•
•
•
•
Xt+1 = RXt
X1 = RX0
X2 = RX1
X2 = R(RX0) = R2X0
Xt = Rt X0
• Taking logs:
• ln(Xt) = t ln(R) + ln(X0) = rt + ln(X0)
where r = ln(R)
Expressed in terms of an intrinsic
growth rate g
• R = (1 + g)
• Xt+1 = (1+g)Xt
• Xt = (1+g)t X0
• Taking logs:
• ln(Xt) = t ln(1+g) + ln(X0) = rt + ln(X0)
where r = ln(1+g)
and when g is small r is approx equal to g
– See Excel file
Simple dynamic patterns
Continuous time exponential growth:
X( t )  X(0) exp( rt )
ln X( t )  ln X(0)  rt
By differentiation, we get the dynamic differential
equation:
dX
 rX
dt
Density Dependent Growth
• Logistic growth as a special case of density
dependent growth
Back to dynamics …
a
N*
b
a. Damped oscillations: stability
N*
b. Constant amplitude
N*
c. Explosive oscillations: instability
d. Chaos
Dynamics
•
•
•
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Stable (periodic) limit cycles
Non periodic solutions
Dependence on initial conditions
Chaos
Modelling approaches
One population of one species: Complete independence
Herbivore (H)
Ht = f(H)
H is all Ht-i for i > 0
One population of one species: Dependent upon a predetermined environment
e.g. Logistic fishery model
Herbivore (H)
Environment (E)
Ht = f(H, E)
H is all Ht-i for i > 0
E = predetermined environment
One population of one species: dependent upon its environment
Two alternative further modelling directions:-
Herbivore (H)
Herbivore (H)
or
Interacting
population (S)
Evolving and/or
stochastic
environment (Et)
Models of species interactions
Forms of interaction - two species (say H and S) are linked
by:
• neutralism
• competition
• mutualism
• commensalism
• amensalism
• parasitism
• predation
Some Wikipedia definitions below
• neutralism: the relationship between two species which do interact but do not affect each
other. True neutralism is extremely unlikely and impossible to prove.
• competition: an interaction between organisms or species, in which the fitness of one is
lowered by the presence of another. Limited supply of at least one resource (such as food, water, and
territory) used by both is required. Examples: cheetahs and lions; tree in a forest.
• mutualism: a biological interaction between individuals of two different species, where each
individual derives a fitness benefit. Example: pollination relationships.
• commensalism: a class of relationship between two organisms where one benefits and the
other is not significantly harmed or benefited. Example: the use of waste food by second animals, like
the carcass eaters that follow hunting animals but wait until they have finished their meal.
• amensalism: one species impeding or restricting the success of the other without being
affected positively or negatively by the presence of the other. Example: black walnut tree, which
secrete juglone, a chemical that harms or kills some species of neighboring plants.
• predation: a biological interaction where a predator (an organism that is hunting) feeds on its
prey, the organism that is attacked.
• parasitism: a type of symbiotic relationship between two different organisms where one
organism, the parasite, takes from the host, sometimes for a prolonged time.
One example of a forms of biological species interaction = predation
(  predator-prey models)
Herbivore prey (H)
Predator (S)