Fig 1.
Schematic illustration of the model.
The central cycle depicts the environmental dynamics where the population alternately experiences the feast phase and the famine phase. Discrete nutrient supply events trigger the transition to the feast phase, whereas nutrient consumption by cells leads to resource depletion, driving the transition to the famine phase. The panels on the right illustrate the two key physiological constraints assumed in the model: (Top) The resource-use trade-off determines the feasible growth rates on distinct nutrients, which defines the spectrum between specialists (orange) and generalists (blue). (Bottom) The growth-death trade-off imposes a cost on fast growth, assuming that phenotypes with higher maximum growth rates () suffer from higher death rates (
) during the famine phase.
Fig 2.
Schematic of the population dynamics model and examples of the simulation with N = 3 and E = 2.
(A) Temporal changes in population size and nutrient levels in a single phenotype (N = 1) case. The blue line represents the population X, and the red dashed line represents the amount of nutrients S. At the time points , nutrients are supplied and S is set to S0. The feast phase (gray shaded area) begins at each nutrient supply event and ends when the supplied nutrient is exhausted by cellular consumption, after which the system enters the famine phase. At each time
, a different type of nutrient is randomly selected and supplied. The type of nutrient supplied depends on the value of environmental variables. (B) The generalist (phenotype 2) is at
; the two specialists (phenotypes 1 and 3) are symmetric at
and
, respectively. (C) The case with
. Two specialists dominate the population. (D) The case with
. The generalist dominates the population. The initial values of populations are set to unity for all three phenotypes. The parameters are set to
, and
. At each nutrient supply event, one of the two nutrient types (A or B) is chosen at random. The waiting time for the nutrient supply
follows a gamma distribution with shape and rate parameters being 2 and 50,
.
Fig 3.
Transition between generalist and specialist dominance.
Temporally averaged populations of the one generalist and two specialists as a function of . Crosses (×) denote results with constant nutrient supply intervals (
), whereas circles (•) denote those with gamma-distributed intervals (
). Above this, the adjacent top panel illustrates the mean duration of the feast phase,
, for the respective conditions. The secondary top x-axis displays the ratio
. Here,
(
) and
(
) denote the arithmetic means of the growth and death rates of the generalist (specialist) across the two environmental conditions. Populations were averaged from t = 2.5 × 105 to t = 3.0 × 105 across 10 simulations; error bars indicate standard error. At each nutrient supply event, one of the two nutrient types (A or B) is chosen at random. All other parameters follow Fig 2C and 2D.
Fig 4.
Temporal average of the population as a function of .
Changes in the temporally averaged populations as is varied in simulation with one generalist and two specialists. The case with
is shown in (A) and the case with
is shown in (B). Above these, the adjacent top panels illustrate the mean duration of the feast phase,
, for the respective conditions. At each nutrient supply event, one of the two nutrient types (A or B) is chosen at random. We fixed the variance (
) and used different distributions with varying means. For each parameter, we ran 10 simulations and averaged the results. The error bars indicate the standard error. All other parameters were identical to those used in Fig 2. The solid lines represent the approximated analytical solutions, with further details on their derivation provided in S1 Text Section 4.
Fig 5.
Normalized population density distribution.
(A) Simulation with two environmental conditions. The x-axis represents the geometric mean of growth rates, and the y-axis represents the difference in logarithmic growth rates between the two environments. We varied from 10−1.0 to 100.3 and ran simulations until t = 2.0 × 105. Population sizes were time-averaged over the last 20 nutrient supply intervals. Growth rates are defined by Eq (14) with m = 2, c = 0.2 and
. (B) Plot of the growth-to-death ratio for each phenotype. The phenotypes with the highest growth-to-death ratio and with the largest population at a given
are highlighted by the solid red line and the dashed blue line, respectively. (C-E) Simulation with three environmental conditions. Each dot represents one phenotype. We simulated the population dynamics up to t = 2.0 × 105. Population sizes were time-averaged over the last 30 nutrient supply intervals. The growth rates of each phenotype across the three environments, denoted as
, were set to
, where x + y + z = 0 holds. The phenotypes are discretely distributed on this plane such that adjacent phenotypes are generated by holding one parameter constant while increasing one of the remaining two by a fixed step size and decreasing the other by the same amount. Through panels A and C to E, the supplied nutrient type is switched deterministically, cycling through the specified nutrient set. For each simulation, the logarithm of the temporally averaged population in each bin, normalized from 0 to 1, is plotted. The parameters are set to a = 0.01, b = 1, S0 = 10 and
. Phenotypes can switch only between adjacent bins, with a transition probability of 10−4.