A new mathematical preprint suggests that the way a trait evolves may depend not only on which trait values reproduction and survival favor, but also on how strongly each process favors them. When those strengths vary across space or time, the study’s models produce effective fitness landscapes that can become asymmetric or develop an incipient second peak—even though the underlying reproduction and survival functions remain Gaussian.
The work examines a basic problem in trait-evolution modeling: reproduction and survival can each have their own preferred trait value, while the combined fitness of an organism reflects both. The authors ask how separating those two components, and allowing their selection strengths to vary, changes the overall landscape and the movement of a population’s mean trait.
The findings come from analytical calculations and computer simulations, with a proof-of-concept fit to previously published data on the plant Chamaecrista fasciculata. They describe what follows from a set of mathematical assumptions; they do not provide a controlled empirical test of whether the proposed patterns occur in nature.
A compromise between two preferred trait values
Under constant selection, the combined reproduction-survival fitness function remains Gaussian, the familiar bell-shaped form used in the model. But its peak is not automatically halfway between the two component optima. The composite optimum is a weighted average, with the stronger stabilizing-selection component pulling the combined peak closer to its own preferred trait value.
In this framework, stabilizing-selection strength describes how tightly each component favors trait values around its optimum. Changing the relative strengths can therefore move the combined optimum without moving either component optimum. The proposed Ornstein-Uhlenbeck-type model makes that connection explicit by linking the effective optimum and the strength of attraction to the same component-specific selection parameters.
The model also predicts a cost when reproduction and survival favor different trait values. The maximum total fitness is lower as the component optima become more separated, with the size of the reduction depending on that separation and on the combined selection strengths.
That result gives the model a simple internal tension: one trait value may be favored by reproduction and another by survival, so the combined landscape represents a compromise rather than a single untouched target. The model’s predicted peak and maximum height are consequently both properties of the two components together.
When selection changes from place to place
The paper then allows selection strengths to differ among locations. Spatial heterogeneity was represented by averaging over distributions of selection strengths and numerically evaluating the resulting drift, the model’s directional tendency for the mean trait to change. With Gamma-distributed strengths, the effective landscape can become skewed and, for some parameter values, begin to show a second mode.
This matters because a population experiencing many local selection regimes need not behave as though it were under one average regime. In the model, the shape of the effective landscape depends on how the strengths of reproduction and survival are distributed across space, not just on the locations of their optima.
The simulations traced how those landscapes affected the distribution of final mean phenotypes. As symmetric heterogeneity increased, the distributions became broader. Under increasing asymmetric heterogeneity, the simulated outcomes shifted toward larger trait values.
The authors used 1000 independent simulations for each specified parameter combination in one simulation set. The setup used a population-size parameter of N = 50, phenotypic variance VP = 1.5, an initial mean phenotype of 6.5 and a simulation horizon of T = 200.
Time variation widened the outcomes
A separate comparison asked whether changing selection through time produced the same pattern as changing it across space. Temporal heterogeneity was modeled by repeatedly updating the selection parameters and solving the resulting stochastic differential equation with the Euler–Maruyama method, a step-by-step numerical approximation.
In that comparison, constant selection and temporal heterogeneity produced similar mean endpoints, but the temporal model produced a broader distribution of endpoints. Spatial heterogeneity, by contrast, shifted the endpoints toward the survival optimum in the specified simulations.
The comparison used 1000 independent simulations, with N = 50, VP = 1.5, an initial mean phenotype of 5.5 and T = 200. The model assigned Gamma distributions with parameters 1 and 5 to δ, and parameters 5 and 5 to γ.
The contrast is one of the study’s clearest modeling results: in these runs, temporal variation was expressed mainly as extra spread around comparable average endpoints, while spatial variation was expressed mainly as a shift in where the long-term outcomes concentrated. Those patterns depend on the parameter distributions used in the simulations.
A plant-data example, with a narrow evidentiary role
To show how the framework could be fitted, the authors used published survival and fecundity data for Chamaecrista fasciculata. They estimated the two components separately with maximum-likelihood models: a Bernoulli likelihood for survival and a normal approximation to a Poisson model for fecundity.
The reported survival estimates were âS = 0.4746, b̂S = 0.0560 and zS∗ = −0.01167. The fecundity estimates were âF = 0.4986, b̂F = 0.00549 and zF∗ = −0.7301, with an overall-fitness parameter β̂ = 148.5948.
Those values are parameter estimates for the proof-of-concept example, not measurements showing that the proposed heterogeneous landscapes operate in the plant population. The authors leave formal model-selection and model-adequacy tests for future work, so the example illustrates how the approach may be used rather than settling whether it outperforms competing explanations.
A framework, not a forecast
The authors present the framework as a possible mechanistic route from variation in selection strength to stochastic models of phenotypic evolution. In practical terms, the model is designed to let the effective optimum and the pull toward that optimum change together when the component-specific strengths change.
That could make the framework relevant to researchers building models of fitness landscapes, stochastic trait evolution, evolutionary rescue or macroevolutionary inference. The paper’s results, however, remain theoretical, analytical and simulation-based, with one proof-of-concept fit to previously published plant data.
The study’s central message is therefore conditional: if reproduction and survival have distinct optima and their stabilizing strengths vary across space or time, the combined landscape and the resulting trait dynamics can look different from the constant-selection case. Establishing whether those assumptions improve explanation or prediction will require formal comparisons and further data.
Paper data and sources
Original title: Novel models of trait evolution via an expansion of Lande's fitness function: The Ornstein-Uhlenbeck process meets the Little Prince's boa
Authors: Jose M Ponciano, Claire Godineau, Laura Jimenez et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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