Modelling concepts Modelling in discrete time difference equations

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: (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 • • 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 X 1 = RX 0 X 2 = RX 1 X 2 = R(RX 0) = R 2 X 0 Xt = R t X 0 • Taking logs: • ln(Xt) = t ln(R) + ln(X 0) = rt + ln(X 0) 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 X 0 • Taking logs: • ln(Xt) = t ln(1+g) + ln(X 0) = rt + ln(X 0) 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: By differentiation, we get the dynamic differential equation:

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 • • 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) 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)