A net reproductive number for periodic matrix models

Figures & data

Table 1. For a Leslie matrix of period p=2, the difference between the characteristic quadratic q(λ) of Q and p(λ), quadratic factor of the characteristic polynomial λ²ⁿ⁻² p(λ) of B, is a multiple k of λ−1. Note that the second factor in k is a sum of non-negative terms and can vanish if and only if each of these terms equals 0.

Dimension n	k
2	φ1 f 1
3
4
5
6
7

Figure 1. Plots of the sensitivities of R ₀ defined by Definition 1 (solid line) or ρ[B] (dashed line) as functions of ϕ₁ in Example 1 are shown for two illustrative cases. (a) For the exponential trade-off function and parameter values ϕ₀=10 and τ₀=0.9, we see that the sensitivities have opposite signs for small values of bad season fertility ϕ₁. This is because lies between the two thresholds−τ*=−0.64855 and−τ**=−0.13222. Note that the sensitivities do have the same sign for larger values of ϕ₁. (b) For the trade-off function and parameter values ϕ₀=5 and τ₀=0.8, we see that the sensitivities have the same signs for small values of bad season fertility ϕ₁. This is because is greater than both thresholds−τ*=−0.5336 and −τ**=−0.1733. Note, however, that the sensitivities do not always have the same signs for all values of ϕ₁.