Department Of Mathematics Maths 208 Assignment 3 Due 4 Pm Tu

Department Of Mathematicsmaths 208 Assignment 3 Due 4pm Tuesday 4 Ju

Analyze a discrete dynamic population model for the rare black robin bird, including transition matrices, eigenvalues, and long-term behavior; examine differential equations related to growth and spreading phenomena with solutions and interpretations; and solve initial value problems related to differential equations modeling disease spread.

Paper For Above instruction

The assignment covers three main mathematical topics: population dynamics modeled through matrix systems, differential equations and their solutions, and applications in epidemiology through differential models of disease spread.

Population Dynamics and Eigen Analysis

The black robin, a critically endangered bird endemic to the Chatham Islands, provides a compelling case study for understanding population dynamics modeled with matrix systems. Initially, the population consists of 30 juvenile chicks and 20 breeding adults, represented as the vector x₀ = (30, 20). The yearly transition of this population is described using a 2×2 matrix,

A =

This matrix encapsulates the reproductive and survival parameters of the species, with specific entries corresponding to birthrates and survival rates.

Interpretation of the Transition Matrix

The entry in the first row, second column (1.75) indicates the average number of chicks produced per breeding adult per year, embodying the reproductive rate of the population. The entry in the second row, first column (0.5) signifies the proportion of chicks surviving to become adults in the next year. The remaining entries show zero contributions, consistent with the model structure.

Calculations for Population Projection and Eigen Analysis

Using the matrix A, the next year's population vector x₁ is obtained by:

x₁ = Ax₀ =