Field synopsis

The aim of this work was to review and analyze mathematical models on adaptations in a changing environment in the relevant fields in biology, investigating their assumptions, parameter spaces and predictions.

Current approaches to study  adaptation in a changing environment

When considering adaptation to a changing environment, various disciplines approach the problem of adaptation differently, as briefly reviewed below. Note that there is no clear distinction between the fields; therefore, it is often a matter of opinion into which field a model, a study, or an approach belongs to.  

Population genetics  

studies changing the genetic composition of populations, focusing on a small number of loci. While population genetics is able to make predictions about how genes evolve and spread throughout large populations, it lacks techniques for handling large numbers of genes. In the context of changing environment, the studies typically focus on 1) a single drastic environmental change and the ability of organisms to survive, adapt rapidly and avoid extinction (evolutionary rescue studies)  [see Schiffers 2013, Uecker 2014, carlson 2014, uecker2016] or  2) random environmental fluctuations, assuming random fluctuations in selection gradients [see Uecker2016, Waxman2011, Peischl2012, Cvijovic2015]. 

Both types of studies look at the establishment of beneficial alleles and investigate the importance of genetic and environmental factors for population survival, including available genetic variation, recombination rates, population structure, and the severity and rate of environmental change. 

Quantitative genetics  

studies inheritance of genetically complex traits, encoded by multiple loci [Pease1989, Lynch1993, Buerger1995, Lande1996]. Quantitative genetics often relies on the infinitesimal model (a large number of loci with small effects) [Fisher1930, Barton2017], assuming fixed variance. In the context of adaptation, the Price equation or its derivatives are often used to estimate the response to selection between generations [Queller2017]. While studies investigating the rate and limits of polygenic adaptation in static landscapes are common [kauffman_towards_1987, gillespie_1983, gillespie_molecular_1984], this approach has been only recently used to study adaptation in changing fitness landscapes [Bell2008, Orr2008, Jain2015, Hollinger2019, Thornton2019, Hayward2019].  These studies often focus on the moving [trait] optimum [kopp2009a, kopp2009b, Matuszewski2014, Matuszewski2015], investigating the critical rate of environmental change beyond which a population declines and goes extinct. 

Ecological models,

in contrast to approaches described above,  often focus on population sizes, life histories, and species interactions. While it is traditionally assumed that ecological and evolutionary timescales are separated, attempts have been made to incorporate rapid adaptation and eco-evolutionary feedback into ecological models, either when considering predator-prey interactions [Hairston2005] or adaptation to climate change [Cotto2017]. 

Adaptive dynamics 

is another mathematical framework focused on eco-evolutionary feedback.  Rather than allele frequencies and genotypes, AD focuses on how the existing trait values develop over time. It investigates the fate of the new, mutated phenotype [hans] and how it affects (and is affected) by the environment. Adaptive dynamics often relies on several simplifications, such as asexual reproduction and rare mutations with small, but not infinitesimal, effects. The central assumption is the timescale separation between ecology and evolution [Lion2018].

Travelling waves

describe the continual increase of population grlwth rate (fitness) due to mutations in a finite population [Hallatschek1783]. This approach, developed by physicists largely independently from biologists [Tsimring1996, Peng2003, Hallatschek2008, Rouyzine2008, Good2012], views the process of adaptation as a wave of fitness (imagine a histogram of individual fitness) travelling along the horizontal axis, increasing the fitness.  Due to the heuristic approach employed by physicists, it may offer unique and important insight into the study of adaptation in the changing environment. Recently, [Nguyen2019] observed travelling waves of fitness experimentally, supporting theoretical predictions. 

Simulations

allow combining many advantages of the approaches described above, removing some of the assumptions and providing additional insight into complex scenarios [Kashtan2007, Matuszewski2015, Cotto2017]. However, simulations must be repeated numerous times in order to provide any useful information. Even with the ever-increasing available computational power and though the computational limits are quickly disappearing, there is still a trade-off between the complexity of the simulated scenario,  number of individuals and number of generations. Furthermore, simulations often offer a limited analytical understanding of the problem and usually focus on a small number of adaptive loci.