**Abstract**

Molting is a crucial and intricate process requiring the timely shedding of an animal’s body part, which may include the epidermis, the pelage, or the entire exoskeleton. The unique mechanisms governing the molting cycles of different species can be quantitatively analyzed to predict patterns and behaviours based on key parameters. From this analysis emerge mathematical models describing the interactions of multiple factors involved in the determination of the optimal molting timing and strategy, the animal’s growth between molts, and the final size of the adult. This report examines the mathematical models of molting patterns and behaviours found in seals, insects, birds, and crustaceans. In Weddell seals, a quantitative analysis of the factors influencing molting allows the development of a model describing the effect of sexual maturity and environmental factors on the molt initiation date. The body size of mature *Manduca sexta *can be predicted from data obtained during its fifth instar and parameters related to growth and time. In birds, a quantitative model describes the optimal molting pattern in relation to migration based on key factors such as energetic cost. Finally, this report discusses the stochastic model representing the growth of crustaceans between molts.

**Introduction**

When studying the natural world, statistical analysis and mathematical models are invaluable tools in identifying patterns and relationships between biological processes. The relevance of quantitative analysis is shown across the animal kingdom, especially when examining the moulting cycle. A connection between the timing of the Weddell seal’s moult, ice sheet behaviour, and the female’s reproductive status becomes clear only through a mathematical analysis. Furthermore, in order to find the influence of external factors on a bird’s molt, a mathematical model can be constructed to describe the ideal relation between migration and molting. By modelling the body size of *Manduca sexta *in relation to time, patterns between the insect’s moult and growth emerge. Subsequently, it is possible to predict a crustacean’s size based on their stochastic growth patterns. These four examples demonstrate how math plays a crucial role in understanding how the moulting process interacts with external conditions and other biological processes. It is clear that both research and industry rely heavily on mathematical analysis to discover and explore patterns and relationships in the mathematical kingdom, both for profit and knowledge’s sake.

**Mathematical Models of Molting Patterns in Weddell Seals**

**Molting Stages of Weddell Seals**

In most mammals, hair structures serve the main purpose of trapping heat and ensuring the mammal stays warm. The Weddell seal, however, has a thick layer of blubber that takes care of thermoregulation, and instead their thick coat of short fur is designed to protect their skin from UV radiation and physical damage from fights. As the year progresses, and the fur is subjected to wind, cold temperatures and ocean water, the seal’s black glossy fur becomes brown and dry and must be replaced in one big moult that typically happens in January and February (Walcott et al., 2020). Weddell seals first moult the fur on the top of their backs, then shed fur in progressively wider stripes. Due to this pattern and the distinct difference between brown old fur and black new fur, it is easy to observe and quantify a Weddell seal’s stage in their moult. In a 2019 study performed by Beltran et al., seals were assigned a value from zero to four to represent their moult stage (Fig. 1).

From the moult codes, they were able to calculate molt duration using the following formula (Fig. 2):

Where *First* and *Last* represent the respective first and last sightings with a given moult code *n*. This data was further analyzed to estimate the moult initiation date of seals that were first observed in moult stages one and above. To back calculate these moult initiation date based on their moult code *k*, the following formula was used (Fig. 3):

Where *First* represents the first sighting with moult code *k,* and Δ*k* represents the mean difference between the start and end of a moult cycle. By subtracting Δ*k* and the sum of the moult durations preceding the study, Beltran et al. (2019) were able to obtain an estimate of the moult initiation date. This data allowed for the creation of mathematical models depicting the moulting cycle of Weddell seals so that the impact of different external factors on the moult cycle could be quantitively analyzed.

**Reproduction and Moulting**

Upon data analysis, it was found that there exists a relationship between a sexually mature female’s moult initiation date and her pregnancy status in year two, with a chi-squared value of 18.9. However, a much stronger relationship exists between year one and year two pupping categories, with a chi-squared value of 130.5. It was determined that “phenology of the intervening molt was indicative of previous reproductive dynamics, but not indicative of subsequent reproductive outcomes” (Beltran et al., 2019). Within sexually mature Weddell seals, males and females that had given birth moulted 16 days later than females that hadn’t been pregnant (Fig. 4).

One potential explanation for this is energetics; females may delay their moult onset to give them time to replenish the energy spent on birth and caring for offspring. However, this later moult could have negative energetic consequences. The average moult initiation date of female seals without offspring is January 6 ± 12 days, which aligns with the warmest air temperature days of the year that occurred January 3-5. Female seals that have reproduced started moulting 16 days later on average, when temperatures are 5°C cooler. In order to moult, seals must reach a minimum temperature threshold in order for their epidermal cells to be able to undergo mitotic division. Therefore, by moulting later females with offspring have to expend more energy to reach this threshold, and have to compensate with foraging. Reproductive status did not appear to influence moult cycle duration (Beltran et al., 2019).

**Ice Phenology and Weddell Seal Biological Cycles**

Seal moult initiation dates from year to year are also closely related to the behaviour of large ice sheets. Every summer, Antarctica experiences ice break outs where large sheets begin to melt and separate. In 2016, the McMurdo Sound ice break out happened 21 days late compared to 2013. In response, the average moult was initiated 10-15 days later. The accepted explanation is that the annual phytoplankton bloom is reliant on pack ice retreat (Beltran et al., 2019). When the ice break out was delayed, phytoplankton growth was stunted which decreased the availability of food for the Weddell seals, delaying their moult (Walcott et al., 2020).

**Mathematical Model of Body Size in Relation to Ecdysis in ***Manduca Sexta*

*Manduca Sexta*

**Growth in the early stages of ecdysis**

*Manduca sexta* (Fig. 5) reaches adulthood after undergoing four molting cycles, which corresponds to five distinct larval instars (Nijhout et al., 2006). The final body size is largely dependent on the last instar, as nearly 90% of the final body mass is accumulated during this phase (Nijhout et al., 2006). In insects, the molting cycle is initiated by the release of the prothoracicotropic hormone (PTTH) and the subsequent synthesis of the ecdysteroid ecdysone (Riddiford, 2009). The release of the PTTH is triggered by internal or external environmental signals, which vary for different insect species. In *Manduca*, the PTTH may only be released during a specific time window at night, called a gate, only if *Manduca* has attained a sufficient size (Riddiford, 2009). During the fifth instar, the secretion of the PPTH is initially inhibited by the juvenile hormone (JH) (Riddiford, 2009). The larva is allowed to feed and grow until it reaches the critical weight, which is the weight at which the JH secretion is gradually inhibited by a rising level of JH esterase (JHE) in the blood (Nijhout et al., 2006). The larva continues to grow until the JH concentration has dropped sufficiently for the PTTH to be released and ecdysone to be synthesised, putting an end to the growth period.

*Quantitative analysis and mathematical model*

Growth is found to be exponential in *Manduca sexta* up until the critical weight is reached, at which point the growth is observed to be gradually declining before coming to an end once ecdysone is released (Fig. 6). Pre-critical-weight growth is described by

{dw\over dt} = k*W (1)

where *W* is the mass, *k* is the growth rate, and *t* is the time (Nijhout et al., 2006). Solving the differential equation (1) gives the following expression for the weight *W* at a time *t*:

W(t) = W_5*e^{k*t} (2)

where *W _{5}*is the initial weight at the fifth instar (Nijhout et al., 2006). Equation (2) adequately describes the growth of

*Manduca sexta*up until the time

*t*, where the critical weight

_{CW}*CW*is reached, which can be derived from (2) (Nijhout et al., 2006):

t_{CW} = \frac{ln(\frac{CW}{W_5})}{k} (3)

Beyond *t _{CW}*, post-critical-weight growth is described by the following differential equations and its associated solution:

{dW\over dt} = k*d*W (4)

W(t) = CWe^{-13*k*e^{-11t}} (5)

where *d* is the rate of decline, found experimentally to be *d*=1.43*e^{(-0.11*}^{t}^{)} (Nijhout et al., 2006) and substituted in equation (5).

Overall, the mathematical model describing the body size of *Manduca sexta* consists of equation (2) and equation (5). Equation (2) is valid up until the critical weight is reached at time *t _{CW}*, which is provided by equation (3). The parameters of the model related to size and growth consist of

*W*, the initial weight at the fifth instar, and the growth rate

_{5}*k*. The critical weight

*CW*is determined from

*W*, as the critical weight is found to be a linear function of

_{5}*W*(Nijhout et al., 2006). The parameters related to time are as follows: the initial time

_{5}*t*at which the growth period begins, the interval to cessation of growth (IGC), the opening time of the gate, and the closing time of the gate (Nijhout et al., 2006). The IGC refers to the time interval between the achievement of the critical weight and the release of the PTTH, which results in the cessation of growth. During this time interval, the growth of

_{0}*Manduca*is described by equation (5). The opening and closing times of the gate influence the final body size of

*Manduca*, as a larva reaching the key size shortly after the gate has closed keeps growing until the gate reopens the subsequent night, which corresponds to an additional 16 hours of growth and 1 to 2 g of weight (Nijhout et al., 2006). All of these parameters may be experimentally measured for a given genetic strain, as the exact values vary across genetic strains and are influenced by environmental conditions (Nijhout et al., 2006). This quantitative description of body size determination thus shows that body size is a system property influenced by the interactions of multiple factors related to growth and time.

**Mathematical Model of Bird Molting Strategies**

**Molting and Migration**

Migration is “a large-scale, seasonal and bidirectional movement of animals” (Barta et al., 2008). Resources are unequally distributed in space and time; therefore, animals need to migrate to have the best possible resources (Barta et al., 2008). Birds, particularly, migrate between a breeding location and another location where they can spend the remaining of the year (Barta et al., 2008). Molting, as another key life event of birds, costs large amount of energy and time. As a result, it may have conflicts with other activities such as reproduction and migration (Barta et al., 2008). Short and highly productive summers are usually used by long-distance bird migrants for reproduction. However, since molting significantly improves feather quality and therefore flight performance, molting often needs to be done in summers before migration as well (Barta et al., 2008). There are two main molting-migration strategies (Ginn & Melville, 1983, Jenni & Winkler, 1994, as cited in Barta et al., 2008): molting can occur in the summer instantly after breeding, or after arrival in the wintering location.

**Model of Optimal Molting Strategies**

Barta et al. (2008) present a model that aims to investigate how bird migrants’ optimal molting-migration strategy can be affected by different factors. Their model considers time, locations, molting and feather quality, energy intake and reserves, migration, and sources of mortality. The time variable t in the model considers the period of a year as 52 weeks. Time t is then t=0, 1,…, 51, with week 0 being the middle of the winter in the Northern Hemisphere. There are four sites in the model: site 1 and site 2 are in the Northern Hemisphere; site 3 and site 4 are in the Southern Hemisphere; site 1 is the most northern and site 4 is the most southern.

Birds’ activities are first modelled with baseline parameters; baselines parameters are parameters such that breeding occurs at site 1 (Barta et al., 2008). The results in both the summer molting scenario and the winter molting scenario are shown in Fig. 7 and Fig. 8, respectively. In the summer molting scenario, all birds both breed and molt on site 1 and molting takes place after breeding; a small portion of birds (10.2%) migrate during molting from site 1 and complete molting on site 2 (Barta et al., 2008). In the winter molting scenario, large portion of birds breed on site 1 but molt on site 4; however, some birds (5.9%) molt on site 1 and about a quarter of them (22.1%) breed on site 4 (Barta et al., 2008).

Birds’ activities are then modelled with changes in food distribution. Changes in food distribution affect food availability at four sites and therefore will affect energy intake and reserves (Barta et al., 2008). This can potentially alter birds’ molting-migration strategy. It has been found that when food is highly available in summer, summer molting strategy (molting on the breeding location instantly after breeding) is used; winter molting (molting after the arrival in the wintering location) is used when food is only highly available for a short time in summer while food is highly available in winter at different locations (Barta et al., 2008). The model also investigates the stability of two molting-migration strategies by changing parameters one at a time. These parameters are concerned with general life history, cost of molting, feather quality and cost of migration (Barta et al., 2008). Barta et al. (2008) discovered that in the summer molting scenario, molting is only influenced by the cost of molting in terms of energy; low cost in energy stimulates birds to molt at site 4, leading to biannual molting of many birds.

**Mathematical Modelling of Crustacean Growth**

**Crustecean and Panulirus ornatus Growth**

Predicting and modelling lobster (*Panulirus ornatus*) growth is an integral part of fishery management. The growth models developed for fish populations assume that growth happens in a continuous manner (Foo, 2020). However, since crustaceans molt, they undergo periods of sudden and rapid growth followed by slower ones. The model applied to crustaceans therefore follows a discontinuous growth trajectory (Foo, 2020). Crustacean growth is stochastic, meaning that it can be “well described by a random probability distribution” (The Editors of Encyclopaedia Britannica, 2011). In other words, stochasticity relates to an approach one can take to model randomness (The Editors of Encyclopaedia Britannica, 2011). The function for crustaceans’ growth is also assumed to be “always increasing or remaining constant and never decreasing” (Merriam-Webster, n.d). Lobster growth, because of their molting periods, can be plotted by a stepwise function. However, this approach is not helpful in determining lobster growth at one specific time.

**Stochastic Modelling**

Two parameters characterize the stochastic model, the molting time interval, and the difference in lobster length between two moults. Specifically, a Lévy process was used to model this situation (Foo, 2020). The Lévy process is often used in everything from calculating insurance risk to telecommunications to laser cooling (Papapantoleon, 2008). Though Lévy processes rely on high-level probability mathematics, an example often used to visualize the process is that of “time analog of a random walk” (Ken-Iti, 1999). This model uses a specific version of the Lévy process called the Gamma process. The following function represents the mean lobster growth between two moults (Foo, 2020).

As for modeling the mean time interval of the molting period, it follows this lognormal distribution.

To test this model, data collected on 39 female and 36 male lobsters over the course of four years was used. The jagged increasing growth of the lobster can be called stepwise increasing (Fig. 9) (Foo, 2020).

Using R, a statistical programming software, Monte Carlo simulations were employed blend or convolute the function for mean lobster growth between two moults and for the mean molting time interval (Foo, 2020). Monte Carlo simulations repeat a random process a very large number of times and store the data of these repeated experiments to help determine a certain probability (White, 2021). “Overall, both males and females displayed a monotonically increasing pattern and converged to a common 𝐿∞ when the time approached infinity Females possess a higher growth rate and a larger asymptotic length compared to males for the 4-year study” (Fig 10) (Foo, 2020).

**Conclusion**

To summarize, mathematics and computer programming are powerful tools for studying the molting cycle of various animals. In Weddell seals, their molting stages can be quantified which allows the creation of mathematical models that can be used to analyze the impact of various factors on their molting cycle. In *Manduca Sexta*, an exponential growth is found, and a separable differential equation is established. With further derivation, weight can be expressed explicitly with respect to time, for both pre-critical-weight growth and post-critical-weight growth. In birds, a model for predicting optimal molting-migration strategy is build based on mathematics knowledge and programming aid: the model can be used to examine how different factors will influence birds’ molting-migration patterns. In crustaceans, high level probability mathematics, along with the help of a statistical programming software, are applied to a model in purpose of addressing the stochastic crustacean growth. The study of these cases hopefully can inspire more analysis on molting in the perspective of mathematics and computer sciences.

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