The Phillip Island penguin parade (a mathematical treatment)

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The Phillip Island penguin parade a mathematical treatment) Serena Dipierro,1, Luca Lombardini 2,3, Pietro Miraglio 2,4, and Enrico Valdinoci 1,2,5,6 Corresponding author 1 School of Mathematics and Statistics,
The Phillip Island penguin parade a mathematical treatment) Serena Dipierro,1, Luca Lombardini 2,3, Pietro Miraglio 2,4, and Enrico Valdinoci 1,2,5,6 Corresponding author 1 School of Mathematics and Statistics, University of Melbourne, Richard Berry Building, Parkville VIC 3010, Australia 2 Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, Milan, Italy arxiv: v2 [] 2 Dec Faculté des Sciences, Université de Picardie Jules Verne, 33, rue Saint-Leu, Amiens CEDEX 1, France 4 Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, Barcelona, Spain 5 Weierstraß Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, Berlin, Germany 6 Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1, Pavia, Italy 2010 Subject Classification: 92B05, 92B25, 37N25. Keywords: Population dynamics, Eudyptula minor, Phillip Island, mathematical models. Abstract Penguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviors in their homecoming, which are interesting to observe and to describe analytically. In this paper, we present a simple mathematical formulation to describe the little penguins parade in Phillip Island. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account natural parameters, such as the eye-sight of the penguins, their cruising speed and the possible fear of animals. On the one hand, this favors the formation of conglomerates of penguins that gather together, but, on the other hand, this may lead to the panic of isolated and exposed individuals. The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behavior of the speed of the penguins, the mathematical treatment to get existence and uniqueness of the solution) is based on a stop-and-go procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home there are also cases in which some penguins freeze due to panic). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguins parade. s: 1 2 1 Introduction The goal of this paper is to provide a simple, but rigorous, mathematical model which describes the formation of groups of penguins on the shore at sunset. The results that we obtain are: The construction of a mathematical model to describe the formation of groups of penguins on the shore and their march towards their burrows; this model is based on systems of ordinary differential equations, with a number of degree of freedom which is variable in time we show that the model admits a unique solution, which needs to be appropriately defined). Some rigorous mathematical results which provide sufficient conditions for a group of penguins to reach the burrows. Some numerical simulations which show that the mathematical model well predicts, at least 1 at a qualitative level, the formation of clusters of penguins and their march towards the burrows; these simulations are easily implemented by images and videos. The methodology used is based on: 1. direct observations on site, strict interactions with experts in biology and penguin ecology, 2. mathematical formulation of the problem and rigorous deductive arguments, and 3. numerical simulations. In this introduction, we will describe the ingredients which lead to the construction of the model, presenting its basic features and also its limitations. Given the interdisciplinary flavor of the subject, it is not possible to completely split the biological discussion from the mathematical formulation, but we can mention that: The main mathematical equation is given in formula 1.1). Before 1.1), the main ingredients coming from live observations are presented. After 1.1), the mathematical quantities involved in the equation are discussed and elucidated. The existence and uniqueness theory for equation 1.1) is presented in Section 2. Some rigorous mathematical results about equation 1.1) are given in Section 3 roughly speaking, these are results which give sufficient conditions on the initial conditions of the system and on the external environment for the successful homecoming of the penguins, and their precise formulation requires the development of the mathematical framework in 1.1)). In Section 4 we present numerics, images and videos which favor the intuition and set the mathematical model of 1.1) into a concrete framework, which is easily comparable with the real-world phenomenon. Before that, we think that it is important to describe our experience of the penguins parade in Phillip Island, both to allow the reader which is not familiar with the event to concretely take part in it, and to describe some peculiar environmental aspects which are crucial to understand our description for instance, the weather in Phillip Island is completely different from the Antarctic one, so many of our considerations are meant to be limited to this particular habitat) also, our personal experience in this bio-mathematical adventure is a crucial point, in our opinion, to describe how scientific curiosity can trigger academic activities. The reader who is not interested in the description of the penguins parade in Phillip Island can skip this part and go directly to Subsection It would be desirable to have empirical data about the formation of penguins clusters on the shore and their movements, in order to compare and adapt the model to experimental data and possibly give a quantitative description of concrete scenarios. 3 1.1 Description of the penguins parade An extraordinary event in the state of Victoria, Australia, consists in the march of the little penguins whose scientific name is Eudyptula minor ) who live in Phillip Island. At sunset, when it gets too dark for the little penguins to hunt their food in the sea, they come out to return to their homes which are small cavities in the terrain, that are located at some dozens of meters from the water edge). As foreigners in Australia, our first touristic trip in the neighborhoods of Melbourne consisted in a one-day excursion to Phillip Island, enjoying the presence of wallabies, koalas and kangaroos, visiting some farms during the trip, walking on the spectacular empty beaches of the coast and cherry on top being delighted by the show of the little penguins parade. Though at that moment we were astonished by the poetry of the natural exhibition of the penguins, later on, driving back to Melbourne in the middle of the night, we started thinking back to what we saw and attempted to understand the parade from a rational, and not only emotional, point of view yet we believe that the rational approach was not diminishing but rather enhancing the sense of our intense experience). What follows is indeed the mathematical description that came out of the observations on site at Phillip Island, enriched by the scientific discussions we later had with penguin ecologists. 1.2 Observed behaviors of penguins in the parade By watching the penguins parade in Phillip Island, it seemed to us that some simple features appeared in the very unusual pattern followed by the little penguins: Little penguins have the strong tendency to gather together in a sufficiently large number before starting their march home. They have the tendency to march on a straight line, compactly arranged in a cluster, or group. To make this group, they will move back and forth, waiting for other fellows or even going back to the sea if no other mate is around. If, by chance or by mistake, a little penguin remains isolated, s)he can panic 2 and this fact can lead to a complete 3 freeze. 2 In this paper, the use of terminology such as panic has to be intended in a strictly mathematical sense: namely, in the equation that we propose, there is a term which makes the velocity stop. The use of the word panic is due to the fact that this interruption in the penguin s movement is not due to physical impediments, but rather to the fact that no other penguin is in a sufficiently small neighborhood. Notice that it may be the case that a penguin in panic is not really in danger; simply, at a mathematical level, a quantified version of the notion of isolation leads the penguin to stop. For this reason, we think that the word panic is rather suggestive and actually sufficiently appropriate to describe a psychological attitude of the animal which could also turn out to be not necessarily convenient from the purely rational point of view: indeed, a penguin in panic risks to remain more exposed than a penguin that keeps moving hence panic seems to be a good word to describe psychological uneasiness, with or without direct cause, that produces hysterical or irrational behavior). It would be desirable to have further non-invasive tests to measure how the situation that we describe by the word panic is felt by the penguin at an emotional level at the moment, we are not aware of experiments like this in the literature). Also, it would be highly desirable to have some precise experiments to determine how many penguins do not manage to return to their burrows within a certain time after dusk and stay either in the water or in the vicinity of the shore. On the one hand, in our opinion, it is likely that rigorous experiments on site will demonstrate that the phenomenon for which an isolated penguin stops is rather uncommon in nature and, when it happens, it may be unrelated to emotional feelings such as fear). On the other hand, our model is general enough to take into account the possibility that a penguin stops its march, and, at a quantitative level, we emphasized this panic feature in the pictures of Section 4 to make the situation visible. The reader who does not want to take into account the panic function in the model can just set this function to be identically equal to 1 the mathematical formulation of this remark will be given in footnote 7). In this particular case, our model will still exhibit the formation of groups of penguins moving together. 3 Though no experimental test has been run on this phenomena, in the parade that we have seen live it indeed happened that one little penguin remained isolated from the others and panic prevailed : even though s)he was absolutely fit and no concrete obstacle was obstructing the motion, s)he got completely stuck for half an hour and the staff of the Nature Park had to go and provide 4 For a short video courtesy of Phillip Island Nature Parks) of the little penguins parade, in which the formation of groups is rather evident, see e.g. The simple features listed above are likely to be a consequence of the morphological structure of the little penguins and of the natural environment. As a matter of fact, little penguins are a marine-terrestrial species. They are highly efficient swimmers but possess a rather inefficient form of locomotion on land indeed, flightless penguins, as the ones in Phillip Island, waddle, more than walk). At dusk, about 80 minutes after sunset according to the data in [13], little penguins return ashore after their fishing activity in the sea. Since their bipedal locomotion is slow and rather goofy at least from the human subjective perception, but also in comparison with the velocity or agility that is well known to be typical of predators in nature), and the easily recognizable countershading of the penguins is likely to make them visible to predators, the transition between the marine and terrestrial environment may be particularly stressful 4 for the penguins see [9]) and this fact is probably related to the formation of penguins groups see e.g. [2]). Thus, in our opinion, the rules that we have listed may be seen as the outcome of the difficulty of the little penguins to perform their transition from a more favorable environment to an habitat in which their morphology turns out to be suboptimal. 1.3 Mathematical formulation To translate into a mathematical framework the simple observations on the penguins behavior that we listed in Subsection 1.2, we propose the following equation: Here, the following notation is used: ṗ i t) = P i pt), wt); t ) ε + V i pt), wt); t ) ) + f p i t), t ). 1.1) The function n : [0, + ) N 0, where N 0 := N \ {0}, is piecewise constant and nonincreasing, namely there exist a possibly finite) sequence 0 = t 0 t 1 t j ... and integers n 1 n j ... such that nt) = n j N 0 for any t t j 1, t j ). At time t 0, there is a set of nt) groups of penguins pt) = p 1 t),..., p nt) t) ). That is, at time t t j 1, t j ) there is a set of n j clusters of penguins pt) = p 1 t),..., p nj t) ). For any i {1,..., nt)}, the coordinate p i t) R represents the position of a group of penguins on the real line: each of these groups contains a certain number of little penguins, and this number is denoted by w i t) N 0. We also consider the array wt) = w 1 t),..., w nt) t) ). We assume that w i is piecewise constant, namely that w i t) = w i,j for any t t j 1, t j ), for some w i,j N 0, namely the number of little penguins in each group remains constant, till the next assistance. We stress again that the fact that the penguin stopped moving did not seem to be caused by any physical impediment as confirmed to us by the Ranger on site), since no extreme environmental condition was occurring, the animal was not underweight, and was able to come out of the water and move effortlessly on the shore autonomously for about 15 meters, before suddenly stopping. 4 At the moment, there seems to be no complete experimental evidence measuring the danger perceived by the penguins, and, of course, the words danger and fear have to be intended for the purposes of this paper, and strictly speaking as a human interpretation, without experimental testing, and thus between quotation marks. Nevertheless, given the swimming ability of the penguins and the environmental conditions, one may well conjecture that an area of high danger for a penguin is the one adjacent to the shore-line, since this is a habitat which provides little or no shelter, and it is also in a regime of reduced visibility. As a matter of fact, we have been told that the Rangers in Phillip Island implemented a control on the presence of the foxes in the proximity of the shore, with the aim of limiting the number of possible predators. Whether the penguins really feel an emotion comparable to what humans call fear or panic is not within the goals of this paper it is of course also possible that, at a neurological level, the behavior of the penguins follows different patterns than human emotions). Nevertheless we use here the words fear and danger to give an easy-to-communicate justification of the mathematical model. Of course, any progress in the study of the emotional behavior of penguins would be highly desirable in this sense. 5 penguins join the group at time t j if, for the sake of simplicity, one wishes to think that initially all the little penguins are separated one from the other, one may also suppose that w i t) = 1 for all i {1,..., n 1 } and t [0, t 1 )). Up to renaming the variables, we suppose that the initial position of the groups is increasing with respect to the index, namely p 1 0) p n1 0). 1.2) The parameter ε 0 represents a drift velocity of the penguins towards their house, which is located 5 at the point H 0, + ). For any i {1,..., nt)}, the quantity V i pt), wt); t ) represents the strategic velocity of the ith group of penguins and it can be considered as a function with domain varying in time i.e. V i, ; t) : R nt) N nt) R, V i, ; t) : R n j N n j R for any t t j 1, t j ), and, for any ρ, w) = ρ 1,..., ρ nt), w 1,..., w nt) ) R nt) N nt), it is of the form V i ρ, w; t ) := 1 µ w i ) ) m i ρ, w; t ) + vµ wi ). 1.3) In this setting, for any ρ, w) = ρ 1,..., ρ nt), w 1,..., w nt) ) R nt) N nt), we have that m i ρ, w; t ) := j {1,...,nt)} sign ρ j ρ i ) w j s ρ i ρ j ), 1.4) where s Lip[0, + )) is nonnegative and nonincreasing and, as usual, we denoted the sign function as 1 if r 0, R r sign r) := 0 if r = 0, 1 if r 0. Also, for any l N, we set µl) := for a fixed κ N, with κ 2, and v ε. { 1 if l κ, 0 if l κ 1, In our framework, the meaning of the strategic velocity of the ith group of penguins is the following: When the group of penguins is too small i.e. it contains less than κ little penguins), then the term involving µ vanishes, thus the strategic velocity reduces to the term given by m i ; this term, in turn, takes into account the position of the other groups of penguins. That is, each penguin is endowed with a eye-sight i.e., the capacity of seeing the other penguins that are sufficiently close to them), which is modeled by the function s for instance, if s is identically equal to 1, then the penguin has a perfect eye-sight ; if sr) = e r2, then the penguin sees close objects much better than distant ones; if s is compactly supported, then the penguin does not see too far objects, etc.). Based on the position of the other mates that s)he sees, the penguin has the tendency to move either forward or backward the more penguins s)he sees ahead, the more s)he 5 For concreteness, if p it ) = H for some T 0, we can set p it) := H for all t T and remove p i from the equation of motion that is, the penguin has safely come back home and s)he can go to sleep. In real life penguins have some social life before going to sleep, but we are not taking this under consideration for the moment. 1.5) 6 is inclined to move forward, the more penguins s)he sees behind, the more s)he is inclined to move backward, and nearby penguins weight more than distant ones, due to the monotonicity of s). This strategic tension coming from the position of the other penguins is encoded by the function m i. When the group of penguins is sufficiently large i.e. it contains at least κ little penguins), then the term involving µ is equal to 1; in this case, the strategic velocity is v that is, when the group of penguins is sufficiently rich in population, its strategy is to move forward with cruising speed equal to v). The function P i pt), wt); t ) represents the panic that the ith group of penguins fears in case of extreme isolation from the rest of the herd. Here, we take d d 0, a nonincreasing 6 function ϕ LipR, [0, 1]), with ϕr) = 1 if r d and ϕr) = 0 if r d, and, for any l N 0, wl) := { 1 if l 2, 0 if l = 1, 1.6) and we take as panic function 7 the function with variable domain i.e. P i, ; t) : R nt) N nt) [0, 1], P i, ; t) : R n j N n j [0, 1] for any t t j 1, t j ), given, for any ρ, w) = ρ 1,..., ρ nt), w 1,..., w nt) ) R nt) N nt), by ) P i ρ, w; t := max {ww i ), max ϕ ρ i ρ j )}. 1.7) j {1,...,nt)} j i The panic function describes the fact that, if the group gets scared, then it has the tendency to suddenly stop. This happens when the group contains only one element i.e., w i = 0) and the other groups are far apart at distance larger than d). Conversely, if the group contains at least two little penguins, or if there is at least another group sufficiently close say at distance smaller than d), then the group is self-confident, namely the function P i pt), wt); t ) is equal to 1 and the total intentional velocity of the group coincides with the strategic velocity. Interestingly, the panic function P i may be independent of the eye-sight function s: namely a little penguin can panic if s)he feels alone and too much exposed, even if s)he can see other little penguins for instance, if s is identically equal to 1, the little penguin always sees the other members of the herd, s
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