Cannibalism and synchrony in seabird egg‐laying behavior

During years of high sea surface temperature, food resources for glaucous‐winged gulls (Larus glaucescens) are scarce. In response, male gulls cannibalize the eggs of neighbors. When this occurs, female gulls in dense areas of the colony adopt a tactic of egg‐laying synchrony, in which they lay eggs synchronously on an every‐other‐day schedule. Field observations show that the first‐laid egg of each clutch is the most likely to be cannibalized. In this paper, we analyzed a discrete‐time model of egg‐laying behavior that tracks egg order in the nest. Using Jury conditions, we found that the equilibrium destabilizes into a two‐cycle as colony density (nests per unit area) increases through a critical value, and that the two‐cycle becomes increasingly synchronous as density increases further. We demonstrated that more eggs survive cannibalism in synchronous colonies than in nonsynchronous colonies.

inversely related to the number of clutch initiations that day. That is, the more first eggs laid on a particular day, the less chance each first egg has of being cannibalized, likely due to predator satiation (Weir et al., 2020).
High levels of egg cannibalism lead to an adaptive strategy of egg-laying synchrony, in which females lay eggs together on an every-other-day schedule. Henson et al. (2010) showed empirically that egg-laying synchrony increases as colony density (number of nests per unit area) increases and is strongest for clutch initiation, meaning gulls tend to lay first eggs synchronously on an every-other-day schedule, which entrains the eggs that follow at 2-day intervals.
These two behavioral consequences of high SST, cannibalism and synchrony, beg the question of colony survival as average temperatures continue to rise. Indeed, SSTs in the North American Pacific Northwest have risen approximately 1°C in the last few decades (Irvine & Crawford, 2011;Strom et al., 2004).
Two previous studies used proof-of-concept models to probe the population-level consequences of synchronous egg laying. Burton & Henson (2014) analyzed the following egg-laying model: x be py y x Here, the time step is one day, x is the number of gulls in the first day of the ovulation cycle, y is the number of gulls in the second day of the ovulation cycle, b > 0 is the inherent number of birds that enter the system each day, e cx − is the probability of the incoming gulls joining the x class, c > 0 represents colony density (number of nests per unit area), p (0, 1)  is the probability that a bird in the y class returns to the x class, and p 1 − is the probability that a bird leaves the system. The value of p controls the expected number of ovulation cycles experienced per female, which is empirically three. Burton & Henson (2014), using c as a bifurcation parameter, proved the existence and uniqueness of an equilibrium solution which bifurcates at a critical value c cr into a two-cycle that becomes increasingly synchronous as colony density c increases.
A major shortcoming in model (1) is that the synchronizing mechanism e cx − depresses the average number of eggs laid per day when c is large. Thus, Burton and Henson were not able to compare the effect of cannibalism on synchronous (c c > cr ) and nonsynchronous (c c < cr ) colonies (Burton & Henson, 2014;Gallos et al., 2018).
To avoid this problem, Gallos et al. (2018) added a preovulation compartment to model (1): Here, w is the number of gulls in the colony that are not yet ovulating. The other parameters and variables retain their meaning from model (1). Gallos et al. showed that model (2) also has a two-cycle bifurcation at a critical value of c and that the two-cycle becomes more synchronous as c increases. In model (2), however, unlike in model (1), the average number of eggs laid per day is constant as a function of c. Hence, it was possible to investigate whether egg-laying synchrony is beneficial to the population in the presence of cannibalism in terms of egg survival, which in fact it was shown to be Gallos et al. (2018). A major shortcoming of model (2) is that it does not track the order in which eggs are laid in a nest. Hence, it is impossible to use this proof-of-concept model to probe the population-level effects of cannibalism and synchrony of first-laid eggs.
In this paper, we modify model (2) by adding state variables to account for egg laying order and limiting the number of ovulation cycles to three, implementing the observation that most gulls lay three eggs per clutch (Henson et al., 2011). Tracking the order in which eggs are laid in the nest allows us to incorporate the cannibalism behavior on first-laid eggs (Weir et al., 2020).
Our egg-laying model, without cannibalism, is Here w is the number of females not yet ovulating, x and y are the number of females in the first and second day of the first ovulation cycle, respectively, z and r are the number of females in the first and second day of the second ovulation cycle, respectively, and s and u are the number of females in the first and second day of the third ovulation cycle, respectively. b > 0 is the inherent number of birds that enter the w class each day when x is small, p (0, 1) 0  is the probability that a bird in the y class will move to the z class, and p (0, 1) 1  is the probability that a bird in the r class will move to the s class. In other words, p 0 is the probability that a bird in the first ovulation cycle will continue on to the second ovulation cycle and p 1 is the probability that a bird in the second cycle will continue on to the third cycle. For simplicity, we assume that each ovulation leads to an egg-laying event, and that the number of first, second, and third-laid eggs in the colony corresponds to x t , z t , and s t , respectively.
In this study, we investigate the dynamics of proof-of-concept model (3) as a function of the colony density c 0  . In Section 2, we investigate the stability of the equilibrium solution, and demonstrate a two-cycle bifurcation at a critical value of c, which corresponds to the onset of egg-laying synchrony. In Section 3, we study the existence and behavior of the bifurcating branches. In Section 4, we incorporate cannibalism of first-laid eggs into the model and compare egg survival in synchronous colonies versus nonsynchronous colonies. This study, although heavily motivated by field work, is a theoretical proof-of-concept investigation that probes the dynamic consequences of the two behaviors of egg cannibalism and synchrony in the context of warming SST and its effect on seabird colonies.

| EQUILIBRIA AND STABILITY
The equilibrium of model (3) is and the Jacobian at the equilibrium is Application of the Jury conditions in Lewis (1977) leads to three conditions for the stability of the equilibrium: If one or more of these conditions suffers a reverse inequality, then the equilibrium is unstable.
Condition (5) is true for all c 0  , and condition (7)   In the next section, we show that the 2-cycle is unique and explore the behavior of its branches as c → ∞.
The values of the 2-cycle branches of model (3) are equilibria of the first composite map (as are the values of the equilibrium branch of model (3)). The equilibrium equations of the first composite map lead to the following equations for w x , , and y: The equilibrium values for z r s , , , and u are easily computed from w x , , and y. From Equations (10) and (9), we obtain Using Equation (10) on the left-hand side of Equation (12) Equating Equations (11) and (14), and writing e cx in terms of y by means of Equation (14), gives c b e y e y be y b e y e y e ye 1 ln Rearranging, we obtain an equation for y:  (14) and (12), respectively. In the next three lemmas, we examine the behavior of dB dy as a function of c (Lemma 3.1 and Lemma 3.2) and then determine the number of roots of B y ( ) as a function of c (Lemma 3.3). Our goal is to show that B y ( ) has exactly one root for c c 1  and exactly three roots for c c > 1 (see Figures 1 and 2).  1. When c c = 1 , V y ( ) has exactly one root which also corresponds to its minimum.
Combining (20) and (21) Thus the root of . Therefore when c c < 1 , , and exactly three roots 1. When c c < 1 , Lemma 3.2 implies dB dy is strictly positive, thus B y ( ) has exactly one root. In the latter case, when c c > 1 , the upper and lower equilibria of the composite map correspond to the values of the two-cycle in model (3), and the middle equilibrium of the composite map corresponds to the (now unstable) equilibrium of model (3). We will now establish that the two-cycle branches approach 0 and b 2 as c approaches ∞ (see Figure 2).

When
Theorem 3.5. The lower branch of equilibria of the first composite map of model (3) approaches 0 and the upper branch approaches b 2 as c → ∞.
Proof. From Equation (22) it is easy to see that tends to infinity. Note also that In summary, the equilibrium of model (3) splits into a two-cycle at c c = 1 . The lower branch of the two-cycle approaches 0 and the upper branch approaches b 2 as c → ∞ (Figure 2). This corresponds to increasing synchrony as c → ∞.

| EFFECT OF EGG CANNIBALISM
We now incorporate cannibalism of first-laid eggs into model (3): Here G > 0 is the number of gull cannibals present in the colony and a > 0 corresponds to the number of first eggs that can be cannibalized per day by each cannibal. For simplicity, we assume that the numbers of first, second, and third eggs laid in a clutch are x z , t t , and s t , respectively. Thus, x aG min ( , ) t represents the number of eggs lost on day t due to cannibalization of the first eggs. Hence, E t represents the total number of eggs that have escaped cannibalization by day t. Note that the inclusion of state variable E does not change the dynamics of the other state variables.
Consider state variable E in model (24) as it depends on a fixed value of c. We denote the solution by E t c and define E t ∞ as the limiting solution when c is arbitrarily large. We want to compare E t c for c c < 1 and E t ∞ to compare the total number of eggs that escape cannibalization in nonsynchronous (c c < 1 ) versus synchronous (c → ∞) colonies. For models (3) and (24) at equilibrium, when c c < 1 , the expected number of eggs laid per day is b bp bp p + + 0 0 1 . In this case, the number of first eggs cannibalized per day is b aG min( , ). For models (3) and (24)  If aG b < 2 , that is, if the number of eggs that can be cannibalized per day is less than the number of first eggs laid every 2 days, then S N > . Thus, on the model attractors, more eggs survive in the synchronous colony. In general, Theorem 4.1. Fix an initial condition vector for model (24) that is independent of c. If aG b < 2 and c c The proof is similar to that of Gallos et al. (2018) and is omitted.

| DISCUSSION
In this paper, we created a model that simulates gull egg-laying habits while tracking the egglaying order. We showed that when the colony density exceeds a certain value c 1 , the system bifurcates and egg-laying synchrony occurs. Furthermore, the system becomes increasingly synchronous as the colony density continues to increase. We implemented the empirical observation that only the first-laid eggs are cannibalized and showed that egg-laying synchrony leads to an advantage in the overall survival of the gull eggs in the presence of cannibalism. Interestingly, although model (3) is more biologically accurate and has more state variables than model (2), the stability criterion which comes from the Jury Conditions is simpler in model (3) than in model (2). In particular, the stability criterion for model (3) has three nontrivial conditions in comparison to four nontrivial conditions for model (2). This is noteworthy because in mathematical modeling an increase in realism usually corresponds to a decrease in tractability. The extra condition for model (2) is due to the "p term" that feeds a percent of the y class back into the x class, where p is set so that the expected number of loops through x and y is 3, which is the expected number of ovulation cycles for a given female. If this "loop" is eliminated in model (2) by setting p = 0, so that all gulls exit the system after class y, the rank of the Jacobian matrix of model (2) is reduced to 2. If two more ovulation cycles (four classes) are appended to the end of the system, with no loops, to account explicitly for all three ovulation cycles as in model (3), the rank remains 2 and the dimension of the nullspace increases by 4. In general, in a model such as model (3), the stability criterion is independent of the number of ovulation cycles. It is therefore not only more realistic, but also more tractable, to model the number of ovulation cycles explicitly rather than via an expected number of circuits through a single loop.
This study is a theoretical proof-of-concept investigation and is not meant to be connected to data, although it is heavily motivated by field data. Biological simplifications include the assumption that the number of birds entering the system has no limit and that the breeding season is infinitely long; these assumptions allow analysis of asymptotic dynamics.
In model (24), the parameter G is the number of gull cannibals present in the colony. This number is essentially constant over a given breeding season and can be estimated directly in the field by counting cannibal territories. Egg cannibals typically carry stolen eggs whole back to their territories to eat them, and these territories are easily identified by the accumulation of fragmented eggshell (Polski et al., 2021). The value of G does depend on SST from season to season, and could also depend on population density, although we have no data on this. Model (24) is an animal behavior model for a single (infinitely long) breeding season, and does not track population numbers across seasons; hence in model (24) we take G to be constant. The population models in Cushing and Henson (2018) and Cushing et al. (2015), however, which are also motivated by the Protection Island system, do incorporate density-dependent cannibalism, and in those studies the authors consider the interaction between density-dependent cannibalism and synchrony.
In summary, in years of high SST, female gulls start to lay eggs synchronously when the spatial density of nests increases through a critical value. The egg-laying becomes increasingly synchronous as density increases further. In the context of a prolonged increase in SST and the resulting cannibalism behavior, specifically first-egg cannibalism, model (24) shows that synchronous colonies produce more eggs than nonsynchronous colonies. This suggests that egg-laying synchrony is beneficial not only at the level of individual fitness, but also at the population level, and may decrease the chance of colony extirpation in the face of climate change.