- Qualification of difference between expected and observed for parameter estimation.  Measure of distance between two vectors.  "Error."

- Often times = 2 (sum of squares). You might use something different is you were interested in the maximum difference, etc.

Advantages: The derivatives are easily found, if stochastic term is normally distributed the sums of squares works the same as other methods.

Disadvantages: Essentially double penalty for large deviations (larger deviations contribute more).

The probability of the observed values given those parameter values.

When using maximum likelihood methods, you want to chose the set of parameters that produce a distribution with the greatest probability of chosing the observed data set.

- Don't have to know joint probabilities (which can be very difficult to figure out).

- Not dependent on sample size

- Easier to calculate error

Integrate over space but time is discrete.  You are looking at a population at different time steps over continuous space.

Includes immigration/emmigration because individuals can be produced at some place y and some get dispersed to x (or could have opposite where they get produced at x and get distributed to y).

μ(t+1, x) = ∫_ΩK(x,y)μ(t,y)dy

μ(t+1,x) - The population at time t+1 at location x.

Ω - The entire domain (space)

K(x,y) - Dispersal kernel.  Governs movement between x and y.

μ(t,y) - Population at time t at place y.  Governs dynamics of areas that feed (or take away from) x.

Individuals live in one place but forage in the surrounding areas (often communal animals).

Modeling approaches:

- Discrete time reproductive events

- Spatially distributed resources

- Foraging strategies in space and time

Resource: f_(t+1) = G(f_t)(1-P(c_t))
- f(_t+1) = food resources at time t+1

- G(f_t) = Recruitment function for consumers.  (Beverton-Holt model for density dependence)

- 1-P(c_t) = Fraction of resource not eaten by consumer

Total consumption: e_t = G(f_t)P(c_t)

- Like the first function but is the amount consumed rather than the amount left over.

Consumer: C_t+1 = S_cC_t + βe_t

- c_t is the population of consumers at time t

- S_c is survival of consumers to the next time step

- β is the conversion coefficient (efficiency of conversion of resources to more crickets).

e_t = ∫G(f_t(x))P((c_t)k(x))dx

- Integrate between -L/2 and L/2 across patch from left to right.  Central place is at 0 so patch is divided in both directions.

- Recruitment is at place x (as a function of food resources).

- Resource consumption is based on the kernel dependency.

∫k(x)dx = 1-S_c/β

Integrate between -L/2 and L/2.  Consumer extinction if S_c+β < 1.  Need a patch size large enough to insure that those parameters stay above 1.

Include a tunable parameter, α (variance), that changes the dispersal kernel.

For each time step, pick α that maximizes total consumption.

As patch size increases, the best adaptive technique is to alternate between different kernels.  Periodic cycling.

Time varies between continuous and discrete dynamics.  Governed by two equations.

dn/dt = f(n,t), t ≠ τ_k

n(τ⁺) = F(n(τ_k)),(τ_k))

F(•) = Impulse (or pulse)

Growing/nongrowing season

Data collection is partially continuous, partially discrete.

dn/dt = rN(N/A - 1) + σ, t ≠ kT/n

N(kT⁺) = N(kT/n) + σT/n

Includes an allelle effects (A).  There is a carrying capacity but we are essentially ignoring it because it is very big.

σ - Background rate of immigration.  σT is immigration during 1 year.

k - year counter.

n - Number of dispersal events per year.

Immigration only occurs during 'dispersal season'

What σ is necessary to rescue a population from the allee effect (pulsed immigration model)?  How does this compare to a continuous model?

For semi-discrete, σ > (n/T)Atanh(rT/4n)

For all positive r, this is larger than rA/4 (which is the requirement for continuous systems).

Pulsed immigration makes it easier for the population to jump above the allee thresehold.

A group of two or more equations that describe the dynamics of a biological system.

1) Nicholson Bailey host parasitoid model

2) Lotka-Volterra predator prey equations

A hybrid dynamical system uses semi-discrete equations.

N = S + I + R

Total population size doesn't change.  Interested in how disease progresses through population.

Susceptible: dS/dt = -βIS

Infected: dI/dt = βIS - νI

Removed: dR/dt = νI

β - contact rate

ν - Recovery rate (or death rate)

The number of secondary infections caused by a single infected individual. 

R₀ = Nβ/ν

S > ν/β

The susceptible population has to be greater than the ratio between the recovery rate and the contact rate.

S > v - r/β

Disease spreads easier in a growing population than in a static population.

v > (b - d)

Disease induced death is greater than b - d.  Disease dynamics puts bounds on the population.  Buffers population from getting too small or too big.

When population is low, disease not spread much.  Pop growth is increases.  When population is high, disease spreads a lot, population declines.

Interested in parameter values but if they are based on specific units, the units could be influencing the dynamics.  Want to get away from using the units to define the parameter.

Start by defining the parameter as two terms: a number and dimensions.

Singlet probabilities are the probability that a cell is either occupied or unoccupied.  Pair frequencies are the probability that a neighbor pair are in the same state.

Conditional pair frequencies are the probability that a neighbor is in a certain state given the state of the focal site.

They are parameters of the partial derivative Jacobian matrix that are used to determine the trajectories of a 2 species model.  β is based on the partial derivation of one species model with relation to x and the other with relation to y. 

γ uses the product of the left diagonal minus the product of the right diagonal.

In order to have a steady state, β is negative and γ is positive.

Elliptoid, degenerate system.

You system always goes back to the initial conditions at some point but initial conditions influence where you end up.  Not all starting points converge on the same node.

dN_i/dt = N_i[r_i + ∑ aij*Nj]

aij - Per capita interaction strength of species j on species i.

Species i could be self-limiting if aii is negative.

s√mc <1

Average interaction strength times the square root of the number of species*the connectance less the one.  Stability with 1) weak interactions, 2) small number of species, 3) not much interactions

θ = 2J_mν

J_{m =} Metacommunity capacity

V is speciation rate

Shape of metacommunity relative abundance distribution

dp/dt = -∂/∂x*(J(x,t)) + σ(x,y)

Rate of particle creation = rate of change with respect to space (1/space)*the rate of particle movement (space * particles/time) + the rate of creation/destruction of particles (particles/time).