Question

Recruits = 0, Juveniles = 21, Non-flowering adults = 77, Flowering adults = 44

1. Multiply the Projection Matrix found in Figure 4 by the Population Vector above to estimate the number of each size class for one year into the future.

2. Calculate the rate of total population growth (R).

3. Identify which classes increased, decreased, or stayed the same in size from the previous year to the projected year.

Projection Matrix P. nigrispinus Pop. Vector (time t) Population Vector (time t + 1) 0 10 0 FFA-R (0 x nR0 x n 0 x nNFFx nFA) (GR-J X nRP nj 0 x nNF 0 x nFA) (0 x nRGN n, PNF X nNFGFA-NF X NFA) (0 x nRGJNF X n, + GNF-FA X NNF PFA X nFA) GR-J 0 PJ 0 0 GJNF PNF GFA-NF 0 GJFA GNF-FA PFa nFA Figure 3. Procedure to multiply a Projection Matrix by a population vector. The initial population vector includes the population size (ni) for each stage-class at time t. From the P. nigrispinus example, stage-classes include: Recruits (R), Juveniles (J), Non-flowering adults (NF), and Flowering adults (FA). Multiplying the Projection Matrix by the Population Vector at time t results in the Population Vector at time t+1 0 0 0.039 2 3 0.044 0.072 0 0 20 2 0 0.436 0.832 0.006 89 83 75 0 0.003 0.006 0.832 63 Figure 4. Example calculation to test your calculations. Fractions on the right side of the equals sign are rounded to whole numbers because they represent individuals X

Answer 1

1. On multiplying the matrix, we get:

0*0 + 0*21 + 0*77 + 0.039*44 = 1.716 ~ 2

0.044*0 + 0.072*21 + 0*77 + 0*44 = 1.512 ~ 2

0*0 + 0.436*21 + 0.832*77 + 0.006*44 = 73.484 ~ 73

0*0 + 0.003*21 + 0.006*77 + 0.832*44 = 37.133 ~ 37

So Population vector (N_t+1) for time = t + 1 is

2

2

73

37

2. Now, the Population growth rate is the product of N_t+1 * N_t^-1

N_t^-1 = [0 0.0025283 0.0092704 0.0052974]

Population growth rate = 0.8778

3. Recruits class increased from 0 to 2 in the projected year

Juveniles class decreased from 21 to 2 in the projected year

Non-flowering adults class decreased from 77 to 73 in the projected year

Flowering adults class decreased from 44 to 37 in the projected year