Question

You are a conservation biologist, and you discover a small population of plants that live in rocky patches of land on the western side of the Superstition mountains (population A). These plants are biennial, meaning that they reproduce only in their second year of life and then die. The population you discover in Year 1 consists of 150 seeds, 30 one-year-old (vegetative) plants, and 10 two-year-old (flowering) plants. Other biologists at ASU have already done the long and hard research of assessing the life history of this plant to complete the demographic matrix (shown below) Based on this demographic matrix, you have some questions about whether this population of plants is expected to grow or decline. Seeds 1 yr 2 yr 10 0.6 M= 0.2 0 0 0 0.6 According to this matrix, in the next year (Year 2) how many of the following do you expect will exist in the population? Show your work. 20. Seeds? 21. One-year old plants? 22. Two-year old plants? 23. Adding up all individuals in the population in (including seeds, one-year old plants, and two- year old plants), do you expect the population of this plant species to grow, shrink, or stay the same from Year 1 to Year 2? 24. If all goes as expected what would be the discrete growth rate of this species from Year 1 to Year 2 (calculate )? 5

Answer 1

We are given a population with 3 age classes.

Now let us see how the age structure of the population changes from year to year.

We form a "population" vector where x,y,z respectively denote the number of seeds, one year old plants and 2 year old plants.

y

Of the total number of seeds present, only a fraction will survive. The other 2 age categories may reproduce and increase the number of seeds. So the new number of seeds is , where a denotes the fraction of seeds that survive and b, c the number of new seeds produced by an individual in the year 1 and year 2 categories respectively.

arby + cz

In this example, individuals aged 1 year cannot reproduce, so b = 0.

The individuals aged 1 year in the next generation can only come from the seeds of this generation. So the new number of one year old plants is
y= P1+0y 02
, where
P1
is the probability that the seed will survive the next year.

Similarly, individuals aged 2 years can only come from the present group of one year old plants. The new number of 2 year old plants is
0xp2y 02
, where
P2
is the probability that the one year old plants will survive the next year.

The population vector for the next year / generation is

ax bycz P1x0y0 0rp2y0z a C Pi 0 0 0 P2 0 y y

The 3 x 3 matrix we found above is the demographic matrix, and we see that  multiplying the demographic matrix and the current population vector gives us the population vector for the next year. This new vector gives us the number of individuals of each age class in the next generation.

Note that this is only a model of the population. We expect the new population vector to be as above, but in a real life situation, this might not happen exactly.

With this in mind, we can do the given problem.

The current age distribution of the population is 150 seeds, 30 1 year old and 10 2 year old plants. Written as a vector, this is .

150 30 10

The age distribution in the next generation can be found by multiplying the demographic matrix by the above population vector.

0.6 0 0.2 0 (150 x 0.6) 100 150 x 0.2 150 190 0 30 30 0 0.6 0 10 30 x 0.6 18 10

We have found that in year 2, there are 190 seeds, 30 individuals aged 1 year and 18 individuals aged 2 years.

20. 190 seeds

21. 30 one year old plants

22. 18 two year old plants

23. For this part, we add up all the individuals to find the population size in years 1 and 2.

Population in year 1 (current population): 150 + 30 + 10 = 190

Population in year 2 (expected population, according to our matrix model): 190 + 30 + 18 = 238

The population of this plant species is expected to grow from year 1 to year 2.

24. Now let us assume that the plant population followed our matrix model precisely. So the population in year 2 is 238, up from the 190 individuals in the previous year.

The discrete growth rate from year 1 to year 2 is the constant $\lambda$ such that
N2 N1
, where
M, N2
are the population sizes for year 1 and year 2 respectively.

This can be found as  

N2 238 1.25 190 N1