3.1. Antigenic Evolution and Immune Memory

First, we present an application of the dual landscape concept to the niche construction that occurs as pathogens influence the state of immune memory in their hosts and, as a result, undergo evolution in their antigens through the process of immune escape. In this example, the phenotype P is the antigen state of the pathogen, and the environment E is the immune memory state of the host population. The pathogen and host populations are each assumed to be monomorphic. Let w be the underlying fitness of a pathogen strain that has never been encountered in a host population. Let there be a fitness cost to the pathogen of being recognized by host immune memory; this cost is greatest when the strain is exactly the same as what the population previously encountered and smaller when the strain is antigenically similar but not identical to the previous strain. If σ is the degree of cross-immunity, the fitness function might be

(4)$\begin{array}{}& {\varphi }_P=w(1-e^{-{(P-E)}^2/{\sigma }^2}),\end{array}$

which has a low-fitness trough where P and E are similar. This adaptive landscape models how pathogens escape the host immune response via mutation. The host environment is the state of immune memory in the population, and the niche constructive landscape is highest for states “matching” the pathogen. This function might be

(5)$\begin{array}{}& {\varphi }_E={\mathrm{ve}}^{-{(P-E)}^2/{\rho }^2},\end{array}$

where v is an underlying benefit to the host through the action of the immune system and ρ is a factor describing cross-immunity from the host point of view. This constructive landscape has a ridge where P and E are similar, due to the advantage to the host of acquiring memory to recognize P. The forms of these models are similar to the many-strain models that appear in the pathogen evolution literature (Gog and Grenfell Reference Gog and Grenfell2002).

The two surfaces shown in figure 2 are actually computed from the model here, as is the vector field in figure 3. (For simplicity, here and for subsequent examples we assume that $\mu (P,E)=\lambda (P,E)=1$ .) Here, niche construction leads to an arms race in which the pathogen evolves by escaping the immune memory of the host, and the host immune memory responds by developing immunity against new strains of the pathogen. In this never-ending process the pathogen never reaches its adaptive peak because the environment keeps shifting away under its own action. As described for figure 1B this describes an organism that climbs a gravel-covered hill that gives way with each step. An alternative but still loose metaphor is that the organism here is a surfer riding a wave: gravity pulls the surfer in one direction while the force of the wave pushes the board in another. The action of these separate forces on the same object is analogous to the two evolutionary forces acting on organisms in dual landscapes.

3.2. Ameliorating the Environment by Degrading Toxins

Suppose the environment of a bacterial population contains a toxin or antibiotic. Niche constructing bacteria can degrade the antibiotic, thereby improving their environment. The variable P (where $0\le P\le 1$ ) tracks this ability, while E ($0\le E\le 1$ ) tracks the quality of the environment, so that higher values of E correspond to higher rates of detoxification and lower concentrations of the antibiotic. We assume this ability comes with a fitness cost c. To outcompete non-niche constructors that would benefit from the improved environment without suffering the cost, niche constructors must be able to monopolize the improvement to some extent (Krakauer, Page, and Erwin Reference Krakauer, Page and Erwin2009; Conlin, Chandler, and Kerr Reference Conlin, Chandler and Kerr2014). Let m be the viability cost incurred by nondetoxifiers when the environment is in the most toxic state, and let w be a constant that scales the fitness. The parameter m can also be seen as the ability of detoxifiers to monopolize the improved environment they help to create. The fitness of bacteria is then

(6)$\begin{array}{}& {\varphi }_P=w(1-\mathrm{cP})(1-(1-E\left)\right(1-P\left)m\right)\text{.}\end{array}$

The bacteria improve the environment in proportion to the niche constructing ability P, and we assume a diminishing return to the environment so that only small improvements can be made to an already very clean environment. Additionally, let the environment be deteriorated by the influx of toxins at a rate δ per unit E. The constructive landscape function can thus be

(7)$\begin{array}{}& {\varphi }_E=\frac{2q\mathrm{PE}}{1+E}-\frac{1}{2}\delta E^2,\end{array}$

where q is the maximum environmental quality that can be achieved when $\delta =0$ .

A vector field corresponding to this model is shown in the right panel of figure 4. Here, the environment is detoxified by the organism, but this improvement is countered by an influx of toxin from external sources when the organism is a weak detoxifier. For trajectories that cross the null isocline, bacteria with a weaker ability to degrade are favored over more efficient degraders that bear a higher fitness cost of this ability. The trajectories reach the maximum value of E and then approach the equilibrium, which is at the intersection of the $\mathrm{dP}/\mathrm{dt}=0$ null isocline (dashed curve) and the dE/dt = 0 null isocline (solid curve). This equilibrium, which is an evolutionarily stable strategy for the organism, is revealed by the dual landscape.

Figure 4.

Dual landscape and vector field of a model of the evolution of microbes that improve their environment by degrading a toxin or antibiotic. Left two panels, adaptive and constructive landscapes with component vectors. Right, vector field in which the components are combined. Dashed curve, null isocline along which $\mathrm{dP}/\mathrm{dt}=0$ ; solid curve, null isocline along which $\mathrm{dE}/\mathrm{dt}=0$ . Parameters are $w=2,m=0.5,c=0.2,q=1,\delta =0.4$ .

3.3. Drug Resistance and Human Behavior

Boni and Feldman (Reference Boni and Feldman2005) modeled the effect of antibiotic drug resistance evolution on human behavior in seeking drug treatment. They identified a feedback through this niche construction process, which leads to complex dynamics. Their microevolutionary model followed allele frequencies in a population. We present a simplified and mesoevolutionary version of this basic idea to show how we can understand it in terms of the dual landscape.

Here, E is the proportion of people who take antibiotics when they are infected with a pathogen. Let P now represent the degree of resistance of a pathogen to a drug. Assume there is a maximum cost c of resistance that applies to the transmission rate of the pathogen that is $(1-\mathrm{cP})\beta$ in the absence of the drug. The recovery rate of the host is γ, and the cure rate due to the drug (i.e., the efficacy of the drug) is ε if the infection involves a drug-sensitive pathogen and in general $(1-P)\epsilon$ .

In the absence of treatment and drug resistance, the fitness of the pathogen is given by its basic reproduction number $\beta /\gamma$ . With the possibility of drugs in the environment, the expected fitness of a pathogen is

(8)$\begin{array}{}& {\varphi }_P=\frac{E\beta }{\gamma +(1-P)\epsilon }+\frac{(1-E)(1-\mathrm{cP})\beta }{\gamma }\text{.}\end{array}$

The left term gives the fitness component when the antibiotic is used, and the right term gives the fitness component in the absence of the antibiotic.

The constructive landscape is given by the benefit or utility to human hosts of taking or not taking antibiotic drugs. The benefit of taking antibiotics is proportional to the drug sensitivity of the pathogen, $1-P$ . The benefit of not taking drugs is an increasing function of the degree of resistance to the drug, P. The benefit derives from saving money on ineffective drugs and from cultural values that favor restraint in the liberal use of drugs. This restraint stems from understanding that drug use contributes to the rise of resistance. We let this benefit be ηP^k, where k models the reluctance of people to refrain from taking antibiotics and η is a proportionality constant that puts this benefit on the same scale as the benefit of taking drugs. If $k>1$ , people are unwilling to give up the antibiotic unless there is complete resistance, and if $0<k<1$ , people readily refrain from using the antibiotic if the pathogen becomes partially resistant. Putting these benefits together, we have

(9)$\begin{array}{}& {\varphi }_E=E(1-P)+(1-E)\eta P^k\text{.}\end{array}$

The left term is the benefit component when antibiotics are used, and the right term is the benefit component when antibiotics are not used.

A vector field for this model is shown in figure 5. In addition to the vectors, this figure shows the two isoclines along which $d{\varphi }_P/\mathrm{dt}=0$ (dashed curve) and $d{\varphi }_E/\mathrm{dt}=0$ (vertical line). The feedback induced by niche construction leads to oscillation in the dynamics as the system tips from sensitive pathogens and high levels of drug use to resistant pathogens and low levels of drug use and back again. This seesaw-like behavior is the same as that described in figure 1C.

Figure 5.

Dual landscape and vector field of model of niche construction in the evolution of drug resistant pathogens. Left two panels, adaptive and constructive landscapes with component vectors. Right, vector field in which the components are combined. Solid line, null isocline along which $d{\varphi }_E/\mathrm{dt}=0$ ; dashed curve, null isocline along which $d{\varphi }_P/\mathrm{dt}=0$ . Parameters are $\beta =0.2,\gamma =0.15,c=0.2,\epsilon =0.05,\eta =0.8,k=1.5$ .

4.1. A Restoration of Landscapes?

This article seeks to reconcile the concept of the fitness or adaptive landscape with the dynamics of niche construction, which focuses on changes induced by organismic action on environments. In particular, we have sought to make Lewontin’s original mathematical description of niche construction (eq. [1]) not just a heuristic but a concrete framework (eqq. [2] or [3]) for showing how evolutionary feedback can arise. The resulting system can be interpreted according to the dual landscape concept that displays the operation of niche construction at the mesoevolutionary level of multiple successive substitutions (as does the adaptive dynamics approach to evolution). This approach can be applied to a range of niche construction processes that operate at such timescales that can be investigated through a pair of continuous variables describing a phenotype and an environmental variable.

We introduce the dual landscape concept as a partial solution to the problem of dynamic landscapes. An advantage of this concept is that it restores landscapes as static objects, thus making it easier to understand the dynamics of niche construction. While the dual landscape may help us to separate and reintegrate niche construction and adaptation as distinct forces (see fig. 2), it is arguably difficult to see their global action in the phenotype-environment space. The vector field is an equivalent way to present the dynamics, and it may be more useful in that it shows more quantitatively how evolutionary change occurs over the whole space. It also shows how evolutionary feedback occurs due to organism-driven environmental change.

What is a good metaphor for evolution, particularly under the action of niche construction? Even though a vector field may be a more precise representation of a given model, a landscape may have greater intuitive appeal. For example, the visual metaphor of the dual landscape illustrates in a concrete way how the direction of phenotypic evolution can reverse, while climbing up a gradient occurs continuously on the adaptive landscape. The distinction between metaphors and models is not clear-cut (Calcott Reference Calcott2008). Lewontin (Reference Lewontin1963) suggested that metaphors have a didactic role and should be based on assumptions of a model. Because metaphors carry elements that do not reflect reality, he cautions against building models from a metaphor. While metaphors are less precise and sometimes more misleading than mathematical models, they may serve to help us understand those models. In that way, a metaphor such as the adaptive landscape can perhaps be viewed as a model of a model.

4.2. Limitations and Extensions

The dual landscape model has limitations shared with other smooth phenotype-based fitness landscape models. Phenotype occupies a low dimensional continuous space, and movement occurs by small increments between neighboring phenotypic states. In reality multiple phenotypes would be under selection, and movement between states ultimately depends on the underlying genetics. We have had to assume a simple correspondence between genotype and phenotype so that mutation too is a simple process. But the functional relationship between genotype and phenotype is complex. An important complexity is epistasis: the interaction between fitness effects at multiple genetic loci leads to ruggedness in genotype-based landscapes (Kauffman and Levin Reference Kauffman and Levin1987; Nahum et al. Reference Nahum, Godfrey-Smith, Harding, Marcus, Carlson-Stevermer and Kerr2015). Genotype-based landscapes are multidimensional, with states that are not neatly ordered as in phenotype-based landscapes. When recombination is possible, movement on the landscape is not as straightforward as it is under incremental mutation steps. In particular, large steps on the landscape are possible (Cooper and Kerr Reference Cooper and Kerr2016). Phenotype landscapes therefore conceal some potentially important details at the genetic level.

Nevertheless, simple phenotype-based landscapes play an important role in strengthening our understanding of complex evolutionary processes. Furthermore, while the dual landscapes would be impossible to draw for higher dimensional phenotypes, the mathematical model could be extended readily. A future development could be to extend and generalize the dual landscape concept to genetic models.

The dual landscape model can be extended in other ways. Coevolution between species is recognized to be an important instance of niche construction in which each species is a component of the other species’ environment (Odling-Smee et al. Reference Odling-Smee, Laland and Feldman2003). In the examples we presented here, we left out coevolution, which is already well studied. Instead, we considered processes in which the environment is abiotic (the presence of a toxin or antibiotic) or biotic but changes through nongenetic processes (adaptive host immune response or host behavior). The notion of a dual landscape can be applied to coevolution in which the application is clearly a natural one: each species has its own adaptive landscape, which is a function of the environment that includes the other species (see also Kauffman and Johnsen Reference Kauffman and Johnsen1991; Doebeli Reference Doebeli, Svensson and Calsbeek2012). In principle, the idea can be extended to multiple species and entire ecosystems. It is difficult to actually imagine multiple surfaces over many dimensions, but this objection applies equally to the case of a single fitness landscape over a space of multidimensional traits.

Another extension of our model is to consider landscapes as functions of allele frequencies rather than phenotypes. The interpretation would be different since the variables would model microevolutionary change in allele frequencies over shorter timescales instead of mesoevolutionary change in phenotypes over longer timescales. Furthermore, the landscapes indicate directions of change but no longer describe fitness. The two landscapes would, however, be smooth surfaces with corresponding vector fields. To outline this idea, in appendix A we describe a simple model of two interacting species—each forming the environment of the other species—following frequencies of two alleles in each population.

4.3. Unified Landscapes and Fisher’s Fundamental Theorem

In general there is no unified landscape to describe niche construction. That is, in a given system it is usually not possible to find a single landscape to capture the dynamical behavior of the system. This is because there is no single surface that behaves like a potential function in physics, where vector fields are conservative and integrals around closed curves equal zero, which applies, for example, to systems with energy conservation. While the parallels between evolution and statistical physics have been noted (e.g., by Walsh, Lewens, and Ariew [Reference Walsh, Lewens and Ariew2002] and more formally by Sella and Hirsh [Reference Sella and Hirsh2005]), for niche construction and adaptation there is in general no potential function unless

(10)$\begin{array}{}& \frac{\partial }{\partial E}\left(\mu (P,E)\frac{\partial {\varphi }_P}{\partial P}\right)=\frac{\partial }{\partial P}\left(\lambda (P,E)\frac{\partial {\varphi }_E}{\partial E}\right),\end{array}$

which would not hold except by serendipity or artifice.

This feature then imposes an inherent limitation on landscapes as a heuristic tool. There is no single landscape that behaves like a real topographic landscape or an energy potential. It also highlights that two distinct forces operate in the evolutionary dynamics we consider here: niche construction and adaptation. We reiterate that the utility of the dual landscape as an aid to intuition is that it separates the two evolutionary forces and reintegrates them in the manner shown in figure 2.

The notion of a unified landscape has connections with Fisher’s fundamental theorem of natural selection (FTNS; Fisher Reference Fisher1999). Under the modern interpretation of the FTNS, the total change in fitness is made of two parts: the change in fitness due to natural selection and the remaining change due to the deterioration of the environment (Price Reference Price1972; Ewens Reference Ewens1989, Reference Ewens2010; Frank and Slatkin Reference Frank and Slatkin1992; Okasha Reference Okasha2008; Frank Reference Frank, Svensson and Calsbeek2012), where “environment” is broadly defined to include some genetic effects beyond additive effects at the gene in question (Price Reference Price1972; Okasha Reference Okasha2008, Reference Okasha2010). In appendix B we show that under a specific interpretation of FTNS the mean fitness ϕ_M(P, E) is satisfied by the differential equations

(11)$\begin{array}{}& \begin{array}{cc}\frac{\mathrm{dP}}{\mathrm{dt}}& =\frac{\partial {\varphi }_M}{\partial E}\\ \frac{\mathrm{dE}}{\mathrm{dt}}& =-\frac{\partial {\varphi }_M}{\partial P}\text{.}\end{array}\end{array}$

Here, there is a single unified surface ϕ_M(P, E) that links environmental deterioration and adaptation.