Model and solve problems that involve real-life situations using matrices, including Dominance and Leslie matrices.

A population of birds is divided into three age classes: juvenile (0-1 year), young adult (1-2 years), and adult (2+ years). The Leslie matrix L models the population dynamics, where each year 40% of juveniles survive to become young adults, 60% of young adults survive to become adults, and 50% of adults survive. Each adult produces an average of 3 juveniles per year, while young adults produce an average of 1 juvenile per year. If the initial population vector is [120, 80, 100]ᵀ (juveniles, young adults, adults respectively), calculate the total population after one year.

Four swimmers, A, B, C and D, compete in a head-to-head tournament where every swimmer races every other swimmer exactly once. The results are: • Swimmer A won against C and D. • Swimmer B won against A and D. • Swimmer C won against B and D. Which of the following is the corresponding dominance matrix D where entry D_{ij} = 1 if swimmer i beat swimmer j, and 0 otherwise?

A species of marsupial has a maximum lifespan of four years. The population is divided into four age groups: juveniles (0–1 years), young adults (1–2 years), mature adults (2–3 years), and old adults (3–4 years). The table provided shows the reproductive and survival data collected for this species. A Leslie matrix L models the population distribution. (a) Construct the Leslie matrix L for this marsupial population. (2 marks) (b) The current population distribution is 800 juveniles, 480 young adults, 288 mature adults, and 101 old adults. Use the Leslie matrix to determine the total population after one year. (2 marks) (c) Determine whether the population is increasing or decreasing by calculating the total population growth rate over this one-year period, correct to one decimal place. (1 mark)

A wildlife reserve monitors a population of rock wallabies across three age groups: juveniles (0–1 year), young adults (1–2 years), and mature adults (2–3 years). Data collected shows that only mature adults breed, producing an average of 3 female offspring per individual. The survival rate from juvenile to young adult is 40%, and from young adult to mature adult is 55%. (a) Construct the Leslie matrix \(L\) for this population model. (1 mark) (b) If the initial population distribution is \(\mathbf{N}_0 = \begin{pmatrix} 120 \\ 45 \\ 18 \end{pmatrix}\) (juveniles, young adults, mature adults), calculate the predicted population in each age group after one year. (2 marks)

A wildlife conservation program monitors a population of native marsupials. The species has a maximum lifespan of four years. Females are classified into four age groups (0–1, 1–2, 2–3, and 3–4 years) with the breeding and survival data shown in the table below. (a) Construct a Leslie matrix \(L\) that models this population distribution. (2 marks) (b) Given an initial population distribution vector \(\mathbf{p}_0 = \begin{bmatrix} 1{,}200 \\ 360 \\ 180 \\ 72 \end{bmatrix}\), determine the total female population in the second year. (2 marks)