Novel models of trait evolution via an expansion of Lande's fitness function: The Ornstein-Uhlenbeck process meets the Little Prince's boa

Abstract: Adaptive topographies form the foundation for much of our understanding of evolutionary change. Lande's 1976 influential paper on the adaptive topography of phenotypes demonstrated how the concept is inherent in both phenotypic and genetic models of evolution, and how the concept can be used to test evolutionary hypotheses given data. Here, we revisit and generalize Lande's original derivation of an equation analogous to Wright's genotypic adaptive topography to the case of two fitness components. A move to two fitness components yields novel predictions about the shape and mechanistic underpinnings of the adaptive topography. The optimum of this updated fitness function is a weighted average of the optima of the two fitness components, with weights given by the relative strengths of stabilizing selection on each component. Temporal or spatial heterogeneity in the strengths of selection for each fitness component create novel shapes (asymmetry, bi-modality, or lack thereof) of the overall fitness function, a possibility demonstrated with a case-study from the published literature. Finally, when combined with Lande's approach to generate an Ornstein-Uhlenbeck (OU) model for the evolution of the average phenotype, our fitness formulation leads to a previously unrecognized family of stochastic differential equation models of trait evolution. These results provide mechanistic justification for non-Gaussian fitness functions (often observed in natural systems), provide a path for testing alternative models generating non-Gaussian fitness functions, and pave the way for future study of the interplay of ecological and evolutionary dynamics, such as in the study of evolutionary rescue.

• The paper derives a dual-component fitness model in which the effective trait optimum is a selection-strength-weighted average of survival and reproductive optima, while discordant optima reduce maximum fitness.
• The paper shows that spatial or temporal variation in selection strengths can create asymmetric, flattened, or bimodal effective landscapes from Gaussian components, with spatial variation shifting outcomes and temporal variation widening them.
• The paper develops diffusion models in which the OU optimum and attraction strength are mechanistically coupled, improving interpretation of likelihood ridges while highlighting identifiability challenges and the need for component-specific data.

Overview

This paper revisits the foundational derivation by Lande (1976) of an adaptive topography for quantitative phenotypes and its associated Ornstein–Uhlenbeck (OU) model of mean-trait evolution. The central move is to decouple fitness into two components—reproduction and survival—each with its own Gaussian stabilizing-selection kernel, characterized by a component-specific optimum ($z_F^\star$, $z_S^\star$) and strength of stabilizing selection ($\delta$, $\gamma$). This decomposition yields three principal results: (i) under constant selection strengths, the composite fitness function remains Gaussian, but its optimum is a weighted average of component optima with weights equal to relative selection strengths; (ii) allowing spatial or temporal heterogeneity in those strengths generates effective fitness functions that are generally non-Gaussian, asymmetric, and potentially bimodal; and (iii) inserting these landscapes into Lande's diffusion derivation produces a previously unrecognized family of stochastic differential equation (SDE) models in which the OU optimum $\theta$ and the strength of attraction $\alpha$ are structurally coupled through the same parameters. The paper supports these results analytically, via simulation, and with a proof-of-concept fit to published data on Chamaecrista fasciculata (2608.20232).

Background: Wright's and Lande's adaptive topographies

The authors first restate the classical framework. Wright's genotypic adaptive topography relates allele-frequency change to the gradient of log mean fitness, $$\Delta p = \frac{p(1-p)}{2}\frac{\partial \ln \overline{W}}{\partial p}$$. Lande's phenotypic analogue connects the breeder's equation to the derivative of mean fitness: assuming a normal phenotype distribution $p(z,t)$ with variance $\sigma^2$,

$$\Delta \bar{z}(t) = \sigma^2 h^2 \frac{\partial \ln \overline{W}}{\partial \bar{z}(t)},$$

where $z_S^\star$0 is heritability. Lande's Gaussian single-component fitness function $z_S^\star$1 then yields an OU process for the mean phenotype with attraction rate $z_S^\star$2 toward a fixed optimum, later generalized by Hansen (1997) and Butler & King (2004) to a nonzero optimum $z_S^\star$3. The key observation motivating this paper is that this tradition treats the optimum's location and the curvature (selection strength) as independent parameters, without asking what biological processes jointly determine them.

A dual-component fitness function

The authors derive Gaussian kernels for each component from first principles. For reproduction, modeling independent reproductive failure across $z_S^\star$4 individuals gives $z_S^\star$5, and a Taylor expansion of the failure probability $z_S^\star$6 around its minimum $z_S^\star$7 yields a Gaussian kernel with "precision" parameter $z_S^\star$8, interpretable as the strength of stabilizing selection on fecundity. An analogous argument gives a survival kernel with optimum $z_S^\star$9 and strength $\delta$0. Because lifetime fitness is the product of component fitnesses, the composite function is

$\delta$1

with total strength of stabilizing selection $\delta$2 and integrative optimum

$\delta$3

Two consequences follow immediately. First, the composite landscape remains Gaussian when selection strengths are constant, so Lande's quadratic approximation is robust—but its interpretation changes fundamentally: the trait optimum is not a free parameter but is mechanically determined by the relative strengths of selection on the two components. Second, discordance between component optima depresses maximal absolute fitness by the factor $\delta$4, linking landscape elevation to the separation of component optima. While this reduction does not directly affect the expected trajectory of the mean phenotype, it can act indirectly through population size and the efficacy of selection.

Heterogeneity in selection strength generates non-Gaussian landscapes

The paper then treats $\delta$5 and $\delta$6 as random effects, motivated by extensive empirical evidence that nonlinear selection varies across taxa, environments, and years. With Gamma-distributed selection strengths integrated out, each component kernel becomes a generalized Student-$\delta$7 kernel, and their product—an average over heterogeneous environments in the spirit of Levene's (1953) multiple-niche model—is generally asymmetric and can exhibit incipient or full bimodality, even though every local component function is Gaussian. In their illustrative parameterization ($\delta$8), the joint optimum sits at 8.969, displaced from both component optima (3 and 10), with a secondary mode emerging near $\delta$9.

As a proof of concept, the authors fit this model to Etterson's Chamaecrista fasciculata leaf-thickness data, estimating survival and fecundity functions separately (via Bernoulli and Poisson-approximating likelihoods respectively; e.g., $\gamma$0, $\gamma$1 for survival; $\gamma$2, $\gamma$3 for fecundity) and composing them into an overall fitness function whose shape closely resembles the theoretically predicted forms. The authors are explicit that this analysis is a demonstration only: formal model adequacy testing and hypothesis comparison remain undone, and the symmetric structure of the model creates label-switching-type identifiability issues that require labeled component-specific data to avoid.

Consequences for deterministic and stochastic trait dynamics

Substituting the dual-component fitness function into Lande's machinery changes the post-selection mean to

$\gamma$4

and hence the per-generation response becomes

$\gamma$5

Carrying this into the diffusion limit yields the SDE

$\gamma$6

The critical structural difference from the Butler–King OU model is that here the stationary distribution has mean $\gamma$7 and variance $\gamma$8, so the mean and variance of the stationary distribution share a mutual dependency on $\gamma$9. Under the standard OU model these are independent. The authors argue that this coupling offers a mechanistic explanation for the ridged likelihood surfaces between $\theta$0 and $\theta$1 commonly reported in macroevolutionary fitting, and that ignoring it will not resolve estimability problems; they recommend incorporating independently estimated component optima from field or laboratory data whenever possible.

Spatial versus temporal heterogeneity: distinct evolutionary signatures

Using a common diffusion coefficient across all simulations so that differences arise solely from the deterministic component, the authors distinguish two heterogeneity modalities:

• Spatial heterogeneity (Gamma-distributed $\theta$2, $\theta$3 averaged before computing the selection gradient) produces a single fixed but non-Gaussian effective landscape. Symmetric heterogeneity broadens the landscape into multiple peaks while preserving symmetry about the midpoint of component optima; asymmetric heterogeneity (varying only $\theta$4) shifts the dominant peak toward the survival optimum and skews endpoint distributions of the mean phenotype accordingly.
• Temporal heterogeneity (resampling $\theta$5, $\theta$6 each generation during Euler–Maruyama simulation) makes both the instantaneous optimum $\theta$7 and the attraction strength $\theta$8 time-dependent random variables. Notably, with Gamma-distributed strengths sharing a rate parameter, $\theta$9 is Beta distributed and the total strength is Gamma distributed—a Beta-Gamma decomposition that cleanly separates fluctuations in optimum location from fluctuations in attraction strength.

Simulations show that temporal heterogeneity leaves the mean endpoint phenotype similar to constant selection but substantially broadens the distribution of endpoints, whereas spatial heterogeneity shifts the location of long-term outcomes. In other words, spatial heterogeneity primarily relocates evolutionary outcomes; temporal heterogeneity primarily increases their variability. Both mechanisms can move the effective optimum even when component-specific selective targets remain fixed, providing a process-based account of fitness-seascape dynamics that complements models focused on shifting optima.

Broader implications

The framework bears on several bodies of theory. For the maintenance of phenotypic diversity, broadened and flattened effective landscapes weaken fitness penalties of moderate maladaptation, expanding the range of phenotypes at high fitness; moreover, asymmetric landscapes under mutation-selection balance should cause standing genetic variation to accumulate preferentially on the shallow side of the optimum, biasing future adaptive trajectories (2608.20232). For evolutionary rescue, temporal variation in selection weights constitutes a mechanism of maladaptation fluctuation distinct from—and complementary to—optimum-shift models and from width fluctuations of a single fitness function. For comparative methods, the coupling of effective optima to selection strengths implies that phylogenetic estimates of "optimum shifts" may conflate changing selective targets with changing selection strengths; whether phylogenetic data contain enough information to separate these is left open. The authors also position geometric weighting (weights proportional to selection strengths) against the demographic elasticity weighting of Cotto et al. (2019), suggesting adaptive compromises reflect two interacting processes.

Limitations and open questions

The paper concedes several constraints plainly. The empirical illustration is explicitly a proof of concept without formal model selection or adequacy assessment; the identifiability issues arising from the symmetric product form (analogous to label switching in mixture models) require component-labeled data that are rarely available. The derivations assume discrete generations, normal phenotype distributions, constant fecundity $\alpha$0 (extending to $\alpha$1 is noted as unexplored), and weak selection relative to phenotypic variance. Adding spatio-temporal heterogeneity simultaneously is acknowledged to carry steep estimation costs given known data hunger of even simple OU fitting. Key open questions include: whether phylogenetic comparative data can distinguish effective-optimum shifts driven by selection-strength variation from shifts driven by moving targets; how effective-landscape geometry interacts with dispersal and demography in range-limit and rescue contexts; and how demographic (elasticity-based) and geometric weighting mechanisms combine in temporally varying environments.

Conclusion

By decomposing fitness into survival and reproduction components, this work shows that Lande's Gaussian adaptive topography conceals a structural dependency: the integrative optimum is a selection-strength-weighted average of component optima, and heterogeneity in those strengths generates non-Gaussian, asymmetric, and bimodal effective landscapes from purely Gaussian parts. Embedding these landscapes in Lande's diffusion derivation yields a family of SDE models in which optimum location and attraction strength are coupled, with direct consequences for stationary distributions, likelihood geometry, and macroevolutionary inference. The framework links adaptive-landscape theory, fitness seascapes, life-history evolution, and evolutionary rescue through a single process-level parameterization of selection-strength variation, while leaving statistical identifiability and model-comparison challenges as the immediate obstacles to empirical deployment.