Mathematical Biology and Ecology Flashcards
N(t)
Total population of a species at time t
Give a word equation for the general form of dN/dt
dN/dt = births - deaths + migration = f(N)
Simple model for population growth where Birth and Death are proportional to N
(single stage/species model)
dN/dt = alpha * N - beta * N
alpha - fecundity rate
beta - mortality rate
Another term for logistic growth
Self-limited growth
Define carrying capacity
The maximum population size that can be sustained
Steady states of logistic growth
N=0, N=K (carrying capacity)
What does it mean to have a steady state of a population system?
Population sizes N* at which dN/dt = 0 for all t ⩾ 0
Can be unstable/stable
Define Asymptotic Stability
Stability at a fixed point
Time response of N* (equilibrium population)
The interval of time over which the perturbed population n of the unstable (stable) linearised dynamics increases (decreases) by a factor e
i.e. time response = 1 / |f’(N*)| for t0 → t1
Typical properties of a predation function p(N)
[note that dN/dt = … - p(N) ]
No predation if no prey, p(0) = 0
Rate of predation saturates as population increases, p(N) → constant as N → ∞
For p(N) = hill function (predation model), what do the constants A and B represent?
A ~ determines how quick the model reaches the saturation level
B ~ saturation level
How to Non-dimensionalise a Population system
U = αN
T = βt
Find dU/dT = dU/dN * dN/dt * dt/dT
Then choose α and β to cancel/simplify constants in dU/dT, or make it into the required format
Collect constants to define new parameters (should result in less parameters now)
Describe Hysteresis
The change in behaviour of the system depends on whether a parameter is increased or decreased. The system remembers where it was. ‘jumping’ effect
Define bistability
The co-existence of 2 steady states.
Depending on the initial population size, the system to converges to one of the equilibrium as t → ∞.
(In a dynamical system, bistability means the system has two stable equilibrium states. Something that is bistable can be resting in either of two states. In the example of a saddle-node bifurcation diagram, at values of the parameter between the two critical values - corresponding to the saddle node birfuractions, the system has two stable solutions.)
Define a Saddle-Node Bifurcation
Ṅ = f(N,k) , where k is a parameter.
There is a critical value kc of k such that when k = kc, f’(N) = 0 at an equilibrium N.
The bifurcation at k = kc is a saddle-node bifurcation if when k passes through kc, a stable-unstable pair of equilibria are either created or destroyed in the
neighbourhood of N*.
Aims of harvesting
Maximise sustainable yield
whilst
Minimising harvesting effort.
Give an example of a harvesting term for a population model
PROPORTIONAL HARVESTING dN/dt = f(N) = ... - hN h ~ harvesting rate CONSTANT YIELD dN/dt = f(N) = ... - Y0 Y0 ~ constant yield
Yield of harvest
Y(h) = h*N_h
N_h ~ the non-zero equilibrium
Maximal yield is found by determining what h gives the largest value
What determines a failed harvest
Yield = 0
Zero equilibrium is stable
Define a Transcritical Bifurcation
Ṅ = f(N;k), where k is a parameter.
There is a critical value kc of k such that when k = kc, f’(N) = 0 at an equilibrium N.
The bifurcation at k = kc is a transcritical bifurcation if when k passes through kc, a stable-unstable pair of equilibria –one of which is fixed– approach each other and exchange stability in the neighbourhood of N*.
Another term for Time Response
Recovery time
Define relative recovery time between 2 equilibria
TR(Y) / TR(0)
or
TR(h) / TR(0)
Relative Yield
Y / Ymax
What can the relative recovery time be written as a function of?
A function of the relative yield (found by solving the yield equation for h, and writing this in terms of Ymax)
One-dimensional difference models for Discrete Time population models
N_t+1 = f(N_t)
f ~ evolution function (map)
For discrete time population models, what are the general crowding and self-regulation effects?
f having a maximum at some critical population density Nm, with f decreasing for N>Nm.
How to find equilibria of Discrete Time Population models
f(N) = N
The equilibria N* of N_t+1 = f(N_t) are the
population sizes N* such that Nt = N* for all t ⩾ 0. These equilibria satisfy f(N) = N
Linear Discrete Time population model
N_t+1 = r N_t N(t) = N0 r^t (r > 0)
In cobwebbing diagrams, where should the first line from the x-axis (N) go to?
f(N)
NOT the line y=N
What are the conditions for stability in a discrete time population model?
If N_t+1 = f(N_t)
| f’(N) | < 1
How to find the stability of bifurcations in Discrete Time Population models
Look at the second iterate of the map
u_t+2 = f(u_t)
Find fixed points/ period 2 solutions of f
(Note that the equilibrium to the first iterate will be solutions here)
Examine when these equilibria are stable
Name the 3 x models for interacting
populations
PREDATOR-PREY system
COMPETITIVE system
MUTUALISTIC system
In a Predator-Prey system, what is the common notation for the predator and prey populations respectively?
N ~ Prey
P ~ Predator
When is the origin the unique equilibrium of the system
dx/dt = A x
(x is a vector)
A has eigenvalues λ1 and λ2 s.t. λ1λ2 =/= 0
Relation between nullclines and equilibria
Equilibria are located at the intersections of the
nullclines.
Another term for Jacobian matrix
Community matrix
Equilibrium stability: Det(A) < 0
UNSTABLE
Equilibrium stability: Det(A) > 0 & Trace(A) > 0
UNSTABLE
Equilibrium stability: Det(A) > 0 & Trace(A) < 0
STABLE
Nullclines of du/dt = F(u)
Where u is a vector and F is a vector function
F(u) = [ f(u) ; g(u) ] f(u) = 0 and g(u) =0 are the nullclines
Determinant and Trace in terms of eigenvalues
det(A) = λ1λ2 Trace(A) = λ1 + λ2
Breifly summarise the stability of equilibria by real parts of eigenvalues
Stable if Re(λ1), Re(λ2) < 0
Unstable if one of Re(λ1) or Re(λ2) is > 0
If Re(λ1) or Re(λ2) = 0, the stability of the equilibrium is determined by the higher order terms in the Taylor expansion. Parameter values correspond to bifurcations.
What is competitive exclusion?
One of the two competing species goes extinct
Features of a model’s phase portrait
Nullclines (with direction of crossing)
Equilibria and stability
Key co-ordinates, including x,y - intercepts
Arrows for direction of travel in regions (find dx1/dt and dx2/dt > or < 0 to determine direction)
Trajectories
What are enzymes?
Proteins that regulate biochemical reactions.
They bind specific reactants, called substrates, accelerating the rate at which they are converted into reaction product by lowering the activation energy (they are biological catalysts)
Law of Mass Action
The rate of a reaction is proportional to the product of the concentrations of the reactants.
Typical initial conditions of concentrations for reaction kinetics
No initial complex or product
e.g. s(0) = s0, e(0) = e0, c(0) = p(0) = 0,
What is the consequent conclusion of preserving the enzyme concentration in a reaction
The concentraion of the enzyme AND its complex stay constant over time
d/dt(c+e) = constant = e0 + c0
(Note that c0 is often 0 by initial conditions assumption that there is no initial complex)
Quasi-steady state hypothesis.
In general, enzymes are only present in very small quantities compared to their substrates.
ε = e0/s0 ≪ 1
We assume that the initial stage of complex formation is very fast, dv/dT ≫ 1, after which it is essentially at
equilibrium, dv/dt ≈ 0
What are autocatalytic reactions?
reactions in which a chemical is involved in its own production
An example of feedback control
Feedback inhibition
Where a substance indirectly decreases its own production
What does PPM stand for?
Population Projection Model
Examples of biologically/ecologically meaningful components of a population model at time t
age, size or developmental stage
Caswell Notation
right eigenvectors denoted by w
left eigenvectors denoted by v
Primitive Matrix
(A^k)_ij > 0 for some integer k ⩾ 1
A positive square matrix is primitive and a primitive matrix is irreducible
Exception of Perron-Frobenius Theorem
α1 = 0
Another term for population inertia
population momentum
By the Perron-Frobenius Theorem, x(t) can be written in terms of λ, t, x0, v and w. What are w and v referred to in this context?
w ~ stable stage structure
v ~ reproductive value
What is the stoichiometry matrix N in Reaction Kinectics
N = (N_ij)
N_ij = number of molecules of reactant Xi producted/consumed by reaction j
(i.e. imagine labels of Ri reactions for columns and xi reactants for rows)
General format of Stage-structured population dynamics
x(t+1) = L x(t)
L ~ Leslie matrix
X_t+1 = F (X_t)
Describe how the entries of the population projection matrix A relates the classification stages together.
The probability of i → j is denoted by A_ji
Transfer function P for matrix perturbations
The ijth entry of P is the amount one needs to add to the ijth entry of A to achieve the target eigenvalue .
Advantage of using Transfer function analysis over Sensitivity matrix calculations
Transfer function analysis gives EXACT results (no approximations)
What does sensitivity analysis do?
Analyses the effect on the dominant eigenvalue of
ABSOLUTE changes to parameters
What does elasticity analysis do?
the effect on the dominant eigenvalue of
PROPORTIONAL changes to parameters
Hadamard product
element by element matrix multiplication
denoted by ○
(A○B)ij = AijBij.
Stability of time dependent system by dominant eigenvalue
This result is intuitive by the eigenmode expansion
λ < 1 ⇒ STABLE
λ > 1 ⇒ UNSTABLE
Potential sources of error where matrix A doesn’t convey the actual real-life transient dynamics
DISTURBANCE: Perturbation in the initial population.
MEASUREMENT ERROR ON A: A is measured incorrectly, or assumed incorrectly to be time independent
What does the pseudo spectrum measure?
Pseudospectrum measures how close ANY number z in the complex plane is to being an eigenvalue
Quantifies range of eigenvalues obtained by perturbing multiple entries of A
What are c and D in the 1D reaction-diffusion model?
c ~ line concentration
D ~ diffusion coefficient, how fast particles disperse
Define travelling wave (reaction-diffusion section)
A travelling wave is one which moves without change of shape and with constant speed c
spatially homogeneous state
no diffusion
What was the fisher equation originally proposed as a model for?
The spread of a favoured gene in a population
It is an example of COUPLED NONLINEAR REACTION-DIFFUSION EFFECTS
Diffusion as a destabalising mechanism
In the absence of diffusion linearly stable uniform steady states may persist, but the addition of diffusion can cause instability of these steady states that lead to pattern formation.
Spatially uniform steady state of (u0,v0) in Turing mechanisms
f(u0, v0) = g(u0, v0)=0.
In the Turing Machines, what are the functions f and g referred to as?
f describes the reaction kinetics of u
g describes the reaction kinetics of v
For the fisher equation, when do acceptable wave-like solutions exist?
c^2 ≥ 4
c^2 ≥ 4KD
Co-existent steady states
where both points in the equilibrium are > 0
Why is v referred to as the reproductive value?
The components of v give the asymptotic reproductive value of each stage
What names are given to the transient bounds of N(t)
Maximum amplification
Minimum attenuation
