SexOnR–09_MaynardSmith. Modelling the Maynard Smith Paradox (Assignment I)

Maynard Smith's Paradox Modeling Step 1 In the first lecture on the problem of the evolution of sex, the problem, which can be called "Maynard Smith's paradox" (slides 30–32), is explained. This paradox is that the appearance of parthenogenetic females in populations of bisexual organisms should lead to the rapid abandonment of sexual reproduction. Why doesn't this happen? Task 1: Building a Type I Maynard Smith's Paradox Model. Build a model in R that investigates the dynamics of a population consisting of females, males, and clonal individuals (which can be exemplified by parthenogenetic females). The input parameters of this model should be: — the initial composition of the population (the number of individuals of each category separately: females, males, and parthenogens); — the number and composition of offspring from individuals who directly produce them; — the upper limit of the population size. Various mechanisms for reducing the population size are possible (if it exceeds the upper limit); at the first step, the simplest of them can be chosen. In each cycle of this model, reproduction should occur, followed by a reduction in the number of individuals. The output should be the dynamics of the numbers of different forms over time. This model can be deterministic (e.g., a female's offspring is always a pair, half females, half males), or it can be probabilistic. It is more interesting to make the model probabilistic. In this case, it is better to choose a probabilistic rounding mechanism for the population size. The next requirement is not mandatory, but it is still better to build the model according to the typical structure of models in the course. A draft R script can be downloaded here. It is clear that it is best to start with a Type I model, i.e., a model that calculates a single cause-and-effect chain and obtains a certain dynamics of the model population composition under the given conditions. Let's try to describe one of the options for building such a model (i.e., develop a conceptual model). One of two options can be chosen. In the first option, parthenogens are present in the model population from the beginning (but most likely in small numbers). In the second option, there are no parthenogens in the population at the beginning of the simulation, but they can appear in it with a small probability. This option fully corresponds to the available data obtained from the study of real populations of many animal species. The second option is more interesting; let's choose it. Conceptual model of Maynard Smith's paradox (Type I): — the model population consists of three groups of individuals: females (they can be denoted as 1 in the model), males (2), and parthenogens (3); — the initial (α) population size (N) of females (fe) in cycle t can be denoted as αtNfe; similarly, the notation αtNma and αtNpg can be used; — the number of individuals of a certain sex at the beginning of the simulation is an input parameter; these numbers can be denoted as α0Nfe, α0Nma, and α0Npg; — as a result of reproduction, there is a transition from the alpha-population to the beta-population; — females (if males are present in the population) produce offspring consisting of females and males; parthenogens produce offspring consisting of parthenogens; — it should be assumed that with a small probability, a parthenogen may accidentally appear in the female's offspring; — the number of offspring of a certain sex from certain parents is an input parameter; for example, the number of females in the offspring of a female can be denoted as fe o fFe (the first f denotes that the female must be fertilized, and for this, the presence of males is necessary); — other notations for input parameters characterizing fertility: ma o fFe, pg o fFe, pg o Pg; parthenogens do not require males for fertilization; — the beta-population size of each group is calculated as the sum of the alpha-population and the offspring (with probabilistic rounding to integers): βtNpg = αtNpg + αtNfe × pg o fFe + αtNpg × pg o Pg; — the upper population limit (K) is one of the model's input parameters; — if the total beta-population size (βtN = βtNfe + βtNma + βtNpg) exceeds the upper limit, the population size is reduced during the transition to the omega-population (if it does not exceed, the omega-population composition is identical to the beta-composition); — the reduction occurs as follows: each individual survives with a probability of K / βtN and dies with a probability of 1 - K / βtN; — the alpha-population composition (alpha-population size of each group) of the next cycle is determined by the omega-population composition of the previous stage: αt+1Nfe = ωtNfe, αt+1Nma = ωtNma, αt+1Npg = ωtNpg. One of the options for initial parameters and conditions that corresponds to the described conceptual model is as follows.

# I.2. Initial state of the system — початковий стан модельованої системи
al_0_N_fe <- 50 # Початкова чисельність самиць
al_0_N_ma <- 50 # Початкова чисельність самців
al_0_N_pg <- 0 # Початкова чисельність партеногенетиків
# I.3. Parameters — параметри
K <- 100 # Обмеження ємності середовища
fe_o_fFe <- 1.99 # кількість самиць в потомстві фертильної (при наявності самця) самиці
ma_o_fFe <- 2 # кількість самців в потомстві фертильної (при наявності самця) самиці
pg_o_fFe <- 0.01 # кількість партеногенетиків в потомстві фертильної (при наявності самця) самиці
pg_o_Pg <- 4 # кількість партеногенетиків в потомстві партеногенетика
# I.4. Experimental conditions — умови експерименту з моделлю
cycles <- 50 # Кількість робочих циклів моделі

The result of the first task can be the following graph. Step 2 After the model described in task 1 is created, models of Type II and III can be built on its basis. Task 2: Building a Type II Maynard Smith's Paradox Model. Build a model in R that performs a series of iterations (repetitions with the same initial parameters and conditions). The result of the second task can be the following graph. Task 3: Building a Type III Maynard Smith's Paradox Model. Build a model in R that checks at what values of female and parthenogenetic fertility bisexual individuals are displaced by clonal ones, and at what values they are not. Step 3 Under the tested conditions, parthenogens displace bisexual individuals. This happens if: — parthenogens appear in a bisexual population; — the number of offspring from a parthenogen is not less than that from a female (or at least exceeds half the number of offspring from a female); — the mortality of parthenogens and bisexual individuals is the same. Most likely, in many cases, the listed conditions are met. Why then did bisexual individuals not arise? Probably, there is some reason that was not taken into account in our model. We can conditionally call it the "Maynard Smith hidden factor." Let's try to find it! Task 4: Building a conceptual model of a mechanism that prevents the displacement of bisexual individuals by clonal ones. Propose a hypothesis about what exactly maintains bisexual populations and protects bisexual individuals from displacement by clonal ones. The same can be formulated as: assume what the "Maynard Smith hidden factor" is, how does it work! Perhaps the hypothesis you propose can be tested by building an R model (or maybe not...). To try to do this, one should first develop a conceptual model. Propose it!