Figure 1.
Graphs of Declining Vitality with Age
(A) Decline in relative vitality with adult age given by Equation 12. For large x (x = 1), half the new-adult vitality is lost by an early age t1, whereas for small x (x = 0.1), it takes until age t2 to lose half the new-adult vitality. The curve x = 0 shows no senescence.
(B) Decline in total vitality showing delayed-onset senescence, and also the tradeoff between senescence and new-adult fecundity, represented by the new-adult recruitment rate b0(x). The curve x = 0 shows no senescence. Here, the senescence rate is age dependent: , with n = 5, and b0(x) = 1 + x (further detailed in Appendix A of Text S1). In both panels, the nonsenescent mortality rate is μ0 = 0.01.
Figure 2.
Graphs of R(x) as Functions of the Senescence Variable x for Various Initial Recruitment Functions b0(x)
Graphs of b0(x) are shown in the inset panels.
(A) As in Equation 14 with C = 1 and D = μ. In this case, R(x) is monotonically decreasing in x, and the evolutionary optimum is at the nonsenescent state x = 0.
(B) As in Equation 14 with C = 8 and D = μ. In this case, R(x) has a minimum at a positive value of x.
(C) As in Equation 14 with C = 20 and D = μ. In this case, R(x) is monotonically increasing, and the evolutionary optimum is the most extreme compressed life history, x = ∞.
(D) A recruitment function not in the class of Equation 14: . In all cases, ω0 = 1, μ = μ0 + g = 0.1, and R0 = ω0/μ, R∞ = κCω0 with
.
Figure 3.
(A) Relative fecundity functions mt(x) taking the form of Equation 15a at age t for x = 0, 0.1, 0.2, 0.5, 1, and 2, as indicated.
(B) Total adult recruitment functions bt(x) = b0(x)mt(x), with b0(x) = 1 + x.
(C) Survival functions st(x) taking the form of Equation 15b.
(D) Mortality rate functions derived from Equation 15b. The senescence rate is of the age-dependent form
, with n = 5 (further detailed in Appendix A of Text S1). Other parameters are αb = αd = 0.5 and μ = 0.01. All graphs are on the same timescale.
Figure 4.
Simulated Adult Population in Presence of a Low Rate of Extrinsic Adult Mortality
(A) Evolution over time in x (population mean and minimum), showing progressively later onset of vitality loss until most have negligible or zero senescence.
(B) Concurrent reduction in μ0 (mean and minimum) until some have zero aging; the dashed line is the optimum μ0* = 0.017, determined by the Marginal Value Theorem (see Appendix B of Text S1).
(C) Adult population (upper black line) and intrinsic immortals (lower red); the dashed line is the carrying capacity K.
(D) Adult age (mean and maximum) showing extended span for intrinsic immortals. Input parameter values: αb = αd = 0.5; random mutational increments of up to ±0.01 in x and μ0; δt = 0.01; B0 = 10, D0 = 0.3; ω0 = B0μ0/(D0 + μ0); g = 0.001; and b0(x) = ω0(1 + 0.2xe−1/x).