BIO 286 Mathematical Information

Mathematical information

Little to no heritability looks like:

randomly distributed points across a graph, where x is parent trait value vs. y as offspring trait value

Phenotypic plasticity charts: IS plastic

Phenotypic values plotted across different environments, where each line is 1 genotype. Low plasticity is indicated by a small slope, and high plasticity is indicated by a large slope.

How to find frequency of heterozygotes in population

Use Punnett square idea: if AA = p², aa = q², and Aa = pq, 2pq is the frequency of heterozygotes in population

Selection differential equation

S = (avg trait value of breeders in population) - (avg trait value in whole population)

Aka. S = mu_s - mu

Response to selection equation

R = (heritability coeff)*S

aka. predicts how phenotypic trait changes over time

Absolute fitness equation

AF = # of individuals of each genotype after selection / # of individuals of each genotype before selection

Relative Fitness equation

RF = each absolute fitness value / highest absolute fitness value

Selection coefficient equation

s = 1 - fitness

Is the % of individuals of that genotype selected against

Effective population size

number of breeding individuals in idealized population

Censused population

size of population that has actually been counted

Harmonic mean equation

(1/Ne) = 1/t * sum of (1/Ni)

Altruism equation

(coefficient of relationship)*(how many additional offspring produced due to altruism) - cost > 0

Survivorship curve type 1

Low initial mortality, many mammals

Survivorship curve type 2

relatively constant, most birds

Survivorship curve type 3

high initial mortality

n_x value (life table)

number alive at start of study

x value (life table)

age

lx value (life table)

Surviving proportion as fraction of all newborns in original cohort

lx = # at beginning of cohort / # at start

d_x value (life table)

the number dying at each age interval

d_x = n_x - n_x+1

b_x value (life table)

number of offspring born per individual in age class

b_x = number of offspring produced by individual in age class/# alive in age class

L_x value (life table)

Mean # of individuals alive between age class x and x + 1

L_x = (n_x + n_x+1)/2

T_x value (life table)

age classes remaining for individual alive in age class

T_x = sum of L_x

e_x value (life table)

life expectancy of individuals alive at age x

T_x/n_x

R_0 value

net reproductive rate

R_0 = sum of l_x*b_x

If it’s equal to 1: population size is stable

If it’s greater than 1: population size is increasing

If it’s less than 1: population size is decreasing

r value

Intrinsic rate of increase - per capita growth rate

r = ln(R_0)/(T_c) = b - d

b = per capita birth rate

d = per capita death rate

T_c = sum of (x*L_x*b_x)/R_0

Exponential growth equation for overlapping generations with no resource limitations

dN/dt = rN

Applies if no resource limitations and no gene flow (immigration and emigration equal)

Exponential Growth for future generations

Nt = N_0*e^(rt)

Maximum recruitment rate (dN/dt)

is at K/2

Logistic growth equation that accounts for carrying capacity

dN/dt = rN* (K-N)/K

Fixed quota graph

If N is too low, fixed quota drives prey extinct

If N is high enough, population stabilizes at a size below K

Fixed effort graph

Yield decreases with lower prey densities

Low slope = low effort

Steep slope = high effort

Stable where harvest rate = recruitment rate

Type I functional response

consumption rates increase with increasing prey density

Type II functional response

easy to find prey, initially eaten in proportion to density, but handling time/digestion cause plateau

Type III functional response

Difficult to find prey at low density. As density increases, predator efficiency to detect prey increases, but handling time eventually causes plateau

Ingestion efficiency equation

energy available / energy ingested

Assimilation efficiency

energy assimilated / energy ingested

Production efficiency equation

energy in new tissue / energy assimilated