Life histories and demography

A sockeye salmon hatches in a gravel bed in a British Columbia tributary of the Fraser River, spends a year or two in a lake, two years at sea, returns to its natal stream, spawns once and dies. A black bear may live 25 years, first breed at four or five, and raise a small litter every second year. A dandelion sets thousands of seeds within weeks of germinating. These differences in the timing of growth, reproduction and death are called life histories, and they determine how fast populations can grow and how they respond to harvest, habitat loss and conservation. Lesson 4 treated all individuals in a population as identical. This lesson adds age and stage: how ecologists measure survival and reproduction at each age using life tables, how they project populations with matrix models, and how natural selection shapes life histories through trade-offs.

Life tables

A life table summarizes how survival and reproduction change with age. A cohort life table follows a group of individuals born at the same time until all have died, which is practical for short-lived organisms or long-term field studies. A static (or time-specific) life table instead records the ages of individuals alive, or at death, at one time and assumes the population is stable. The standard columns are:

From the life table, three key numbers summarize the population. The net reproductive rate, \( R_0 \), is the average number of female offspring a newborn female produces in her lifetime. The generation time, \( T \), is the average age at which females give birth. The intrinsic rate of increase can then be approximated:

If \( R_0 > 1 \), each female more than replaces herself and the population grows; if \( R_0 < 1 \), it declines.

import numpy as np

# Invented cohort life table for a small mammal (females only)
x  = np.array([0, 1, 2, 3, 4, 5])
n  = np.array([1000, 300, 160, 70, 20, 0])
m  = np.array([0.0, 1.2, 2.6, 2.8, 2.0, 0.0])
lx = n / n[0]
dx = n - np.append(n[1:], 0)
qx = np.divide(dx, n, out=np.zeros(len(n)), where=n > 0)
lm = lx * m
R0 = lm.sum(); T = (x * lm).sum() / R0; r = np.log(R0) / T
print(" x    nx    lx     dx    qx     mx   lxmx")
for row in zip(x, n, lx, dx, qx, m, lm):
    print("{:2d} {:5d} {:6.3f} {:5d} {:6.3f} {:5.1f} {:6.3f}".format(*row))
print(f"R0 = {R0:.3f}  T = {T:.2f} years  r = {r:.3f} per year")

 x    nx    lx     dx    qx     mx   lxmx
 0  1000  1.000   700  0.700   0.0  0.000
 1   300  0.300   140  0.467   1.2  0.360
 2   160  0.160    90  0.562   2.6  0.416
 3    70  0.070    50  0.714   2.8  0.196
 4    20  0.020    20  1.000   2.0  0.040
 5     0  0.000     0  0.000   0.0  0.000
R0 = 1.012  T = 1.92 years  r = 0.006 per year

In this invented population, a newborn female produces almost exactly one daughter over her life (R₀ ≈ 1.01), the average age of mothers is about 1.9 years, and the population is nearly stable, growing at well under 1 percent a year. A small drop in first-year survival would tip it into decline. Note that 70 percent of individuals die in their first year: high juvenile mortality is common in nature, and it means that what happens to the young often controls population growth.

Survivorship curves

Plotting \( l_x \) on a logarithmic scale against age gives a survivorship curve. Raymond Pearl described three idealized types in 1928. Type I curves stay high until old age and then drop, as in humans in wealthy countries and many large mammals that invest heavily in few offspring. Type II curves fall in a straight line on the log scale, meaning a constant mortality rate at every age, as in many songbirds once they leave the nest. Type III curves plunge early and then flatten, as in trees, fish and marine invertebrates that produce huge numbers of tiny offspring, almost all of which die young. Real species mix these shapes, but the types are a useful shorthand.

Matrix population models

Life tables summarize the past; matrix models project the future. Patrick Leslie introduced the approach in 1945. The population is divided into age classes, and a matrix gives the fecundity of each class (top row) and the probability of surviving from one class to the next (sub-diagonal). Multiplying the matrix by the vector of numbers in each class gives the numbers one time step later:

After many time steps, the proportions in each class settle into the stable age distribution, and the population grows by a constant factor each step. That factor is the dominant eigenvalue of the matrix, which equals \( \lambda \), and the stable age distribution is its corresponding eigenvector. Matrix models extend naturally to stages rather than ages, which is often more useful: a tree's fate depends on its size, not its age, and a turtle's on whether it is an egg, a juvenile or an adult.

import numpy as np

A = np.array([[0.0, 1.5, 2.2],
              [0.4, 0.0, 0.0],
              [0.0, 0.6, 0.5]])     # last class: adults that survive and stay adults
n = np.array([100.0, 0.0, 0.0])
for t in range(1, 6):
    n = A @ n
    print(f"year {t}: " + "  ".join(f"{v:7.1f}" for v in n) + f"   total {n.sum():7.1f}")
vals, vecs = np.linalg.eig(A)
i = np.argmax(vals.real)
lam = vals[i].real
w = np.abs(vecs[:, i].real); w = w / w.sum()
print(f"dominant eigenvalue lambda = {lam:.3f}  (r = {np.log(lam):.3f})")
print("stable stage distribution:", np.round(w, 3))

year 1:     0.0     40.0      0.0   total    40.0
year 2:    60.0      0.0     24.0   total    84.0
year 3:    52.8     24.0     12.0   total    88.8
year 4:    62.4     21.1     20.4   total   103.9
year 5:    76.6     25.0     22.9   total   124.4
dominant eigenvalue lambda = 1.175  (r = 0.162)
stable stage distribution: [0.609 0.207 0.184]

Sensitivity and elasticity

Matrix models answer a practical question: if we could improve survival or reproduction at one stage, which would raise \( \lambda \) the most? Elasticity measures the proportional change in \( \lambda \) caused by a proportional change in one matrix element. In a famous analysis in 1987, Deborah Crouse and colleagues built a stage-based model of loggerhead sea turtles in the southeastern United States. Conservation had focused on protecting eggs and nesting beaches, but the model showed that \( \lambda \) was far more sensitive to the survival of large juveniles and subadults, which were drowning in shrimp trawls. This analysis supported the requirement for turtle excluder devices in US shrimp nets. The same logic is now used for species from orcas to orchids.

import numpy as np

A = np.array([[0.0, 1.5, 2.2],
              [0.4, 0.0, 0.0],
              [0.0, 0.6, 0.5]])
vals, W = np.linalg.eig(A); vals_l, V = np.linalg.eig(A.T)
i = np.argmax(vals.real); j = np.argmax(vals_l.real)
lam = vals[i].real
w = np.abs(W[:, i].real); v = np.abs(V[:, j].real)
S = np.outer(v, w) / (v @ w)          # sensitivities
E = S * A / lam                        # elasticities (sum to 1)
names = ["F1", "F2", "F3", "P1", "-", "-", "-", "P2", "P3"]
for (r, c), e in np.ndenumerate(E):
    if A[r, c] > 0:
        print(f"element A[{r},{c}] = {A[r, c]:.1f}   elasticity {e:.3f}")
print("sum of elasticities:", round(E.sum(), 3))

element A[0,1] = 1.5   elasticity 0.145
element A[0,2] = 2.2   elasticity 0.190
element A[1,0] = 0.4   elasticity 0.335
element A[2,1] = 0.6   elasticity 0.190
element A[2,2] = 0.5   elasticity 0.140
sum of elasticities: 1.0

Life history trade-offs

Organisms have limited energy and time, so investing more in one function means investing less in another. These trade-offs shape life histories. A female can produce many small offspring or a few large ones; she can breed early and risk her own survival, or delay breeding and grow; she can breed once in a massive effort or spread reproduction over many seasons. Experiments confirm the costs: when ecologists enlarged clutches of collared flycatchers by adding eggs, parents raised more young that year but survived less well and produced fewer young later.

Breeding once and dying, as Pacific salmon, many annual plants and agave do, is called semelparity. Breeding repeatedly, as most birds, mammals and perennial plants do, is iteroparity. Semelparity is favored when adult survival between breeding attempts is low or when a huge single effort is far more productive than several small ones, as when salmon must make a costly migration to spawn. Interestingly, Atlantic salmon, unlike Pacific species, can survive to spawn again.

In the 1960s, Robert MacArthur and E. O. Wilson described a contrast between r-selected species, favored in unpredictable or newly opened habitats, which mature early and produce many small offspring, and K-selected species, favored in crowded, stable habitats, which mature late, produce few large offspring and invest heavily in each. The scheme proved too simple, and ecologists now usually describe a fast–slow continuum of life histories, with mice, annual weeds and many insects at the fast end and elephants, albatrosses, whales and bristlecone pines at the slow end. Position on the continuum predicts how a species responds to threats: slow species recover slowly from losses of adults, which is why long-lived sharks, sturgeon and rockfish are so vulnerable to overfishing.