Internal cues for optimizing reproduction in a varying environment

Abstract

In varying environments, it is beneficial for organisms to utilize available cues to infer the conditions they may encounter and express potentially favourable traits. However, external cues can be unreliable or too costly to use. We consider an alternative strategy where organisms exploit internal sources of information. Even without sensing environmental cues, their internal states may become correlated with the environment as a result of selection, which then form a memory that helps predict future conditions. To demonstrate the adaptive value of such internal cues in varying environments, we revisit the classic example of seed dormancy in annual plants. Previous studies have considered the germination fraction of seeds and its dependence on environmental cues. In contrast, we consider a model of germination fraction that depends on the seed age, which is an internal state that can serve as a memory. We show that, if the environmental variation has temporal structure, then age-dependent germination fractions will allow the population to have an increased long-term growth rate. The more the organisms can remember through their internal states, the higher the growth rate a population can potentially achieve. Our results suggest experimental ways to infer internal memory and its benefit for adaptation in varying environments.

Keywords: bet-hedging; developmental plasticity; fluctuating environment; life-history evolution; phenotypic variation; population growth.

Figure 1.

(a) Schematic illustration of Cohen’s model of seed dormancy in annual plants. Each year may be good or bad for plant growth. A seed can either germinate to produce a yield $Y_{\epsilon }$ that depends on the environmental condition $\epsilon$, or stay dormant with a probability V of still being viable next year. The number of seeds at the end of year t is N_t. The parameter values used in our calculations are Y₀ = 0, Y₁ = 4 and V = 0.9. (b) The distribution of duration of consecutive good years and bad years. We choose the duration of good years to follow a geometric distribution with a mean of 5, and the duration of bad years to have a Gaussian distribution with a mean and standard deviation of 5 ± 2 cut off at 0 and 10. (c) A state diagram that represents the seed age. Each state $s_{\alpha }$ represents a seed of age α. Blue arrows represent dormancy that increases the age by 1; orange arrows represent germination that may produce new seeds of age 0. Weights on the arrows represent the probability of germination or dormancy.

Figure 2.

The long-term growth rate $\Lambda$ of populations with different sources of information. (a) The value of external cues: organisms can respond to environmental cues through phenotypic plasticity. ${\Lambda }_{\max }$ is the maximum possible growth rate attainable if the population has perfect information about the future environment. ${\Lambda }_{\mathrm{bet}}$ is the highest growth rate achievable by a bet-hedging population without receiving cues, which is suppressed by the entropy of the environment $H(\epsilon )$. ${\Lambda }_{\mathrm{cue}}$ is the growth rate when the population utilizes a cue ξ that has a mutual information $I(\epsilon ;\xi )$ with the environment. (b) The value of internal memory: organisms can utilize their internal states as indirect cues to adjust their behaviour. ${\Lambda }_{\mathrm{bet}}$ from bet-hedging also represents the case with no memory, which corresponds to having only one internal state (L = 1). More internal states (L > 1, see figures 5 and 6) can provide more information about the future environment, $I({\epsilon }_t;{\alpha }_{t-1})$, and allows a higher growth rate ${\Lambda }_{\mathrm{int}}$ for the population. ${\Lambda }_{\mathrm{mem}}$ is the highest growth rate achievable by organisms with a perfect memory (L → ∞) of their lineage history. Memory is beneficial provided that the past environment contains information about the future environment, i.e. $I({\epsilon }_t;{\{{\epsilon }_s\}}_{s<t})>0$.

Figure 3.

Probability of the coming environment ${\epsilon }_t$ conditioned on the seed age α_t−1 at the beginning of year t, as calculated by simulating a lineage of seeds. Dashed line is the marginal probability of the environment, which would indicate that the seed age is uncorrelated with the environment. Blue bars are when the population uses a bet-hedging strategy with a constant germination fraction. Orange bars are when the germination fraction depends on the seed age to maximize population growth rate. In both cases, the seed age is correlated with the environment and thus useful as an internal cue.

Figure 4.

Dependence of the germination fraction q on the seed age α that maximizes the population growth rate. Blue bars are when the environment is temporally structured, as described by the duration of good and bad years in figure 1b. Orange bars are when the environment is drawn independently each year, for which the germination fraction need not depend on seed age and is equal to the bet-hedging solution in equation (2.2) (dashed).

Figure 5.

State diagrams for age-dependent germination. (a) The germination fraction q depends on the seed age α up to α = L − 1, beyond which it remains the same. Varying the length L effectively varies the memory capacity of the organisms. (b) With only one state (L = 1), the organism effectively has no memory, and the germination fraction is a constant, corresponding to simple bet-hedging. (c) The two-state case corresponds to a Markov process where the organisms switch back and forth between two phenotypes, with transition probabilities P(ϕ₁|ϕ₀) = q₁ and P(ϕ₀|ϕ₁) = 1 − q₀.

Figure 6.

Long-term growth rate $\Lambda$ of populations that have different memory capacity as measured by the number of internal states L. For each L, the age-dependent germination fractions $q_{\alpha }$ are chosen to maximize $\Lambda$. Also plotted is the mutual information I between the previous seed age α_t−1 and the environment ${\epsilon }_t$. Both $\Lambda$ and I increase monotonically with the memory capacity L, approaching their respective limits as L ≫ 5 (mean duration of bad years). (Inset) Long-term growth rate $\Lambda$ increases monotonically with the mutual information I. Grey diagonal line represents Cohen’s model with external cues, in which $\Lambda ={\Lambda }_{\mathrm{bet}}+I$.

Figure 7.

The distribution of the duration of consecutive germinations or dormant years along a lineage of seeds. Different colours correspond to age-dependent germination fractions $q_{\alpha }$ for different memory capacities L. (a) For each L, the duration of germinations matches a geometric distribution with a mean of 1/q₀ (dashed line for L = 2 and solid line for L = 10), meaning that there is no memory of previous germinations. (b) The duration of dormancy has a distribution that changes shape depending on the memory capacity L. L = 2 (phenotypic switching) results in a geometric distribution with a mean of 1/(1 − q₁) (dashed line). Larger L results in deviation from a geometric distribution, which is indicative of having internal memory.