Mathematical Modeling for the Life Sciences (Universitext)

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Of course, it is not the unique possibility. The other steady states satisfy: We can easily check that u1 and u3 , when they exist, are stable steady states, and that u2 , when it exists, is an unstable steady state. We start from an initial condition u 0 small. The population converges to the smallest stable steady state u1. The population converges to a new steady state, that we still denote by u1 since it is qualitatively close to the previous state. This solution is now qualitatively close to the previous steady state u3: Now we start from the outbreak steady state u3.


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How can we manage to come back to the refuge steady state u1? Let us look at the graphical resolution again. Unfortunately, we can check that the population size converges to the greatest stable steady state u3 and not to u1. The spruce budworm has not been eradicated. Our system is not reversible: Of course this non-reversibility has a cost in terms of environmental management, from both economical and ecological viewpoints.

Various predation terms may be considered. When a prey species vanishes, the predator changes its strategy and prefers to eat another prey species rather than to spend time and energy to hunt a rare prey. When this species vanishes, the predator still continues to hunt this prey. An example is given in exercise 2. Lotka-Volterra model Model The historical predator-prey model is due to Volterra 2. This model is usually called Lotka-Volterra model since it has been simultaneously introduced by Lotka see [56, 84].

The aim is to explain why one can observe oscillating population sizes3. One of our tasks is to understand whether the oscillations are due to some external cause for instance oscillations of the environment or are due to the internal dynamics of the species.

Mathematical Modeling for the Life Sciences (Universitext)

Let us build the Lotka-Volterra model. Of course these assumptions are very coarse. As usual, the aim is not to build a realistic model, but with few relevant parameters to describe a qualitative behavior, i. Let N t be the number of preys and P t the number of predators. Our assumptions lead to the equations: Volterra began a study of analytical models in order to explain such observations.

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Several examples of oscillating population sizes can be found in the literature e. We eliminate the quadratic terms of the equations. The variables of 2. The steady point 1, 1 is in-between these two intersection points. The trajectories of the Lotka-Volterra system are contained in the curve 2. Average population number We have seen that solutions to Lotka-Volterra are periodic functions.

Let T be the unknown period.

Mathematical Modeling for the Life Sciences

The average number of preys resp. An integration of equation 2. Simultaneous evolution of the numbers of preys and predators 0 1 2 3 Preys 4 5 6 Fig. This is a three levels trophic model. We generalize the Lotka-Volterra model: This is a Lotka-Volterra model indeed. This has been observed, for instance, with rabbits and foxes. The concomitant hunting of rabbits and foxes leads to an explosion of rabbits.

This three levels trophic model can be generalized to multi levels trophic models. When the number of trophic levels is odd, the population size of the lowest level is high; when the number of trophic levels is even, the population size of the lowest level is low4.

The abundance of plankton -and therefore the color of the sea- depends on the parity on the number of trophic levels. We sketch here some outlines of these models e. The following qualitative conclusions can be drawn from the previous assumptions. Its stability matrix is. The extinction of the populations is impossible. This point can either be stable, or unstable.

Either the solutions converge to a steady point, or they converge to a limit cycle. That means that a two-species model will either stabilize steady state case or will be periodic limit cycle case. No other qualitative behavior is allowed. Such an example is given in exercise 2. These species are in competition since they use the same single resource. Can these two species coexist or not? This leads to the concept of ecological niche for species. The modeling is similar to the Lotka-Volterra one and is left to the reader.

A modeling based on the logistic model is the following: We assume 0 cf. Similar computations show that 0, K2 is unstable. There is no other admissible steady state. Both K1 , 0 and 0, K2 are unstable. Now there exists another admissible steady state: To obtain global qualitative results, graphical results e. For dP t dN t and of are easily obtained from a given point N, P , the signs of dt dt equation 2.

We therefore roughly know the direction of the trajectory at this point N, P. This limit can only be a stable steady state. There is no coexistence between the two populations, since their ecological niches are too close. The same graphical resolution is used. There is a coexistence of the two species: The spatial evolution has been neglected.

Our aim is now to introduce spatialtemporal models and to study some simple but usual qualitative phenomena. The global population at time t is Rk N t, x dx. Again we can write a conservation equation: Applying the Divergence Theorem to 2. This is left to the reader. In the most general situation, D is itself a function of N and x. If we assume D to be constant, equation 2. We need to specify an initial condition N 0, x and a boundary condition. In other words, does there exist solutions, are they positive? Moreover we assume that f is Lipschitz: Ronald Fisher, , is mostly known for his statistical works, but he has also worked on genetical and population dynamical models.

Suppose that we know a sub-solution N 1 and a super-solution N 2. If N 1 and N 2 are bounded functions, it is rather easy to obtain a constant C. A little bit of algebra proves that, if Z is negative: Let us apply this result to Fisher equation 2. Assume that the initial population is less than the carrying capacity K, i. Let N be the solution. Solution N can be considered both as a sub-solution and supersolution.

Fisher equation is well-posed and there is no explosion of the population. Suppose that f is linearizable about N0 , i. Let us give an example. The Fisher equation becomes: A linearization about u t, x leads to: The steady solution 1 is stable. Sexual dimorphism is marked at the adult stage: So, this species is poorly moving from a population dynamics point of view, but it is considerably more movable from a genetic population point of view. Measurements of population have been done by counting the number of caterpillars per Bud branch: Experimental data are summed up in Figures 2.

Two phenomena clearly appear. Let us have a look on a given area cf. The dynamics of the Moths is periodic, with a succession of highly elevated peaks, corresponding to the crisis of the pest, and bottoms, corresponding to quiet periods. The period between two peaks is around nine years. This typically is the dynamics of an host-disease model: A propagation phenomenon clearly appears when looking to the spatiotemporal repartition: Let us model the Alpine arc by a onedimensional curve.

The spatial dynamics of the Moth looks like a wave traveling along this curve. The careful study of [4] indicates that this traveling wave is not due to some external factor, but is due to the internal dynamics of the Moth. Let us consider the basic waves: One-dimensional traveling waves We will restrict ourselves to the one-dimensional case: Spatio-temporal evolution of the Larch Bud Moths logarithmic scale 2. Temporal evolution of the Larch Bud Moths in Engiadina logarithmic scale 32 2 Continuous-time dynamical systems 2.

Substituting the traveling wave N into 2. This gives qualitative results on the traveling waves and on the wave speeds. Let us linearize equation 2. We will then study these steady states as usual. For biological reasons11 , function f vanishes at 0: An analogy with mechanics can be done. The Lyapunov function can be viewed as the energy of the system. A dissipative system has a decreasing energy and cannot have any limit cycle.

Function L is monotonic. The trajectory cannot converge to a limit cycle. To sum up, either the trajectory of 2. The steady points are 0, 0 and 1, 0. Point 1, 0 is a saddle point. Let us have an analogy with the steady point. If a steady point is unstable, any perturbation will remove the trajectory from this point: This is the same for a traveling wave: If the linearization is valid, the equation becomes: Dynamik von Laerchenwickler-Populationen laengs des Alpenbogens in German.

Lecture Notes in Biomathematics, Springer-Verlag. Biochemical Oscillations and Cellular Rhythms: Elements of Mathematical Ecology. Qualitative analysis of insect outbreak systems: Spatial patterning of the spruce budworm, J. Let us consider the growth of a mass of bacteria in a Petri dish. The mass grows uniformly in every direction. Only the bacteria on the surface of the mass reproduce. Let N t be the number of bacteria at time t. Justify the following model and solve it: Study the one species model given by: Let us consider the following model: A species N is subject to a seasonal periodic constraint that changes its carrying capacity.

The proposed model is: Consider the logistic model with proportional-rate harvesting: What is lim hT? Consider the one-single species model with predation: Study the model population outbreak, hysteresis with the following predation terms: Consider an insect population N t. A method to eradicate this population consists in introducing sterile insects n t.

These sterile insects prejudice the reproduction of the other insects. The following model is proposed: The parameters a, b and k are positive. Give a condition on n to be sure of the eradication of N t. We assume that the sterile insects are dropped once and that the dynamics of the sterile insects is Malthusian: Let the birth and death rates be constant. Is this solution realistic? Study the steady state and their stability in the model: Consider the predator-prey model: Propose a change of variables such that the new model becomes: This exercise is an extension of the competition model cf.

Find the steady states and their stability. Study analytically the following competition model. The interaction between two species is not always a competition or a predation interaction; the interaction between two species could be to the advantage of both species. The simplest model is: Is this model realistic? We incorporate carrying capacities for both species: In , a rabies pest, coming from East Europe, arrived in France from the east border.

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Foxes were the main vehicle of the rabies. We consider the foxes to be divided into two groups, infective I and susceptible S. You can adopt the point of view of the rabies virus! What is your best strategy? This method is modeled by: Give conditions on the parameters that prevent the rabies pest. A second method to eradicate the rabies pest consist in vaccinating the foxes. Which method do you chose? Consider a disease that, after recovery, confers immunity. The population at time t, denoted by N t , is divided into three groups: The model dynamics is then: The parameters r and a are positive.

Show that N t is constant. Show that the functions S, I and R are positive. Vaccine-impregnated baits that looked like meatballs were dropped by airplane. Give a qualitative interpretation of the previous results. Justify the following dispersion model: Study the travelling waves solutions with boundary conditions: Is such a travelling wave allowed by the Fisher equation? Show that D contains the unstable manifold of the point 0, 1. Conclude and apply to the Fisher equation. Consider the one-dimensional spatio-temporal model of the spruce budworm: Solve the equation Hint: Consider the following model: Justify this delay model from a biological point of view.

The new equation becomes then: What do you think about the stability of point 1? In some biological situations, such hypothesis is not relevant.

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For instance, we can think of the reproduction of some animals or plants which only occur during a short period in the year. In such cases, it will be more relevant to think in terms of discrete time, the time step being equal to a year. The writing of a discrete-time dynamical model does not put any major problem. In fact there is a kind of heuristic parallel between continuous and discrete-time models.

Thus a time wise demographical population will be modeled either with an autonomous differential equation or with a recurrent equation; a population considering age and sexual maturity will be modeled with a Mc Kendrick-Von Foerster-type partial derivative equation or with a discrete delay equation,. We could naively believe that continuous-time and discrete-time models have the same qualitative characteristics.

This will constitute the core of this chapter. We will nevertheless start with some basic reminders on discrete recurrent equations with or without delay, but we will not linger. Before dealing with the study of discrete-time models, let us start with a short reminder of the qualitative characteristics of the continuous-time dynamical systems presented in the previous chapter. Such behaviors are very stable: Except for hypothetically critical values, an alteration of the initial characteristics of the system does not have an impact of the resulting characteristics.

We will focus on the case of an isolated population, taking its temporal evolution into account. Our study-case will be the discrete logistic model derived from the Verhulst continuous logistic model. As its continuous cousin, the discrete logistic model only depends on one parameter.

When this parameter is small, the system converges to a steady point. When it increases, there emerges a cycle of period two, which is new when you consider the continuous logistic equation. Let us go on and increase the parameter of the discrete logistic equation: Now there is no more connection with the continuous logistic model, no longer.

Solutions to the discrete logistic model display astonishing features. In particular, some solutions are very sensitive to changes of the initial condition or to minor alterations of the parameter. Such models are called chaotic models. What about the biological implications of such mathematical results? Can we indeed model a real population with such a sensitive system?

Would it be more realistic to resort to probabilistic modeling? Long-term predictions seem delicate or even impossible with such models. Practically speaking, he has to choose between determinist and random models. Hence, we need an experimental approach which enables us to make such a discrete logistic model valid from a biological viewpoint. That is what [16] did through the study of the dynamics of Tribolium, which we will reproduce after the study of the discrete logistic model. Such bridges will help make their model empirically valid and will enable us to stand in favor of modeling through a chaotic-type model.

It is sometimes more natural, when modeling the evolution of a population, to take into account not only the current situation, but also the past one: We will study the stability of the steady state of these delay models. Indeed, this is a steady point of 3. A linearization about the steady point is done.

A steady state of the system without delay is stable if: The discrete-time logistic model is: From now on, we set: A one-dimensional continuous-time model has no periodic behavior; a discrete-time model can have a periodic behavior. A cycle of period m is a sequence c0 , c1 ,. To every rn , a small interval is associated, for which the cycle of period 2n is stable. Note that a general result due to Sarkovsky [77] ensures that the existence of a cycle of period 3 implies the existence of cycles of period k, with k being an arbitrary integer.

The existence of a cycle of period 3 therefore plays a key-role, for the existence of very disturbed behaviors, called chaotic behaviors [55]. We will only study the particular case: Let us write the initial condition: The sequence un becomes: Cycles of arbitrary periods can be built from a rational initial condition.

Moreover, the convergence 3. Indeed, assume the existence of a stable cycle: This indicates that the population can almost vanish from time to time, but that this population will nevertheless never completely vanish. Assume that the population size is driven by a purely stochastic model, say the sequence un is an i.

The strong law of large number shows that, for every function f: We have seen that an iterative discrete map, even very basic, as in the logistic map, can lead to very complicated behaviors, especially chaotic behaviors. The following question then arises. Is there such mathematical behavior in the biological real world?

The distinction between a chaotic determinist behavior and a stochastic behavior seems particularly delicate to establish empirically. Indeed, a real situation always contains uncertainties: When the model has a stable behavior e. On the other hand, when the behavior is unstable the sensitivity to initial conditions is then essential , such uncertainties play a key-role. This protocol is a way of deciding whether a population dynamics is determinist chaotic or stochastic.

There are three stages in the life of a Tribolium: Wn represents the worms that eat; Pn represents the great worms, the worms that do not eat, the chrysalis and the sexual immature adults; An represents the sexually mature adults. The time unit is two weeks. This is approximatively the time a Tribolium spends in stages Wn and Pn. The rate r is the reproduction rates per adult by time unit, and without cannibalism. One stable steady state. A stable cycle of period 8. Cycle of period 3 together with chaotic behavior and stable cycles of period 8 and more.

Stable cycle of period 3. The transition steady states, cycles and chaos is observed. It seems then reasonable to conclude that Tribolium should have a chaotic behavior. Moreover, an even minor change in these parameters can dramatically modify the dynamics of this species. Last, despite a common belief cf. Chaotic dynamics in an insect population. Quantitative universality for a class of nonlinear transformations.

Period three implies chaos. Simple mathematical models with very complicated dynamics. Bifurcations and dynamic complexity in simple ecological models. Coexistence of cycles of a continuous map of a line into itself in Russian. Compare the stability of the steady states in the following models: For which values of parameter r are there cycles of order 2?

Are the cycles stable? Determine the stability of the possible steady state in the model: Let k be a positive kernel which integral is equal to 1. Assume the following repartition of the population through time: Justify this dispersal model. Calculate the total population size at time n. Now we assume that the kernel k and the function N0 are compactly supported.

What is the colonization speed of R by the population? Now assume a periodic model:: Its life cycle is described by a set of stages i. Let hi be the proportion of stage i surviving the harvest. Let y be the yield of an individual in each stage. The total yield is: We want to maximize the yield per individual. Show that this maximization reduces to the linear programming problem: Find H in terms of the optimal vector u. It is up to the players to choose their strategy in the game.

From a mathematical viewpoint, the players try to make the most of the numerous strategies. Hence the problem of an equilibrium in the game combining each strategy of each player. You can read general books such as [3, 30, 71, 81, 87]. The application of the game theory to the evolution is much more recent than its application to economy [61]. From the point of view of evolution, two issues are commonly dealt with: We chose to thoroughly deal with the hawkdove model. The problem is as follows: The second one is peace: Firstly, we will study the static aspect of the game.

The dynamical aspect is still in progress from a mathematical viewpoint. We will propose a dynamical version of this game, which will allow an interesting parallel with the dynamic demographical models. Finally we will propose exercises as an extension of the game: We consider two players A and B. These players would model either individuals, either populations. These two players are in competition. The aim of player A resp. B is to chose a strategy x resp. For convenience, we denote by A resp. B the set of strategies of player A resp. In the sequel we will give examples of strategies: Let us follow the classical approach.

We assume that the players do not chose their strategies, but only chose the probability of playing a strategy. Therefore we have obtained a continuous set A of strategies. These strategies are called mixed strategies as opposed to the strategies of A that are called pure strategies. A mixed strategy is a convex combination of pure strategies: Such a pair is called a Nash equilibrium of the game. We will therefore admit the results without any proof. Indeed, we will give results on Nash equilibrium in a more general setup than those used in the examples.

The sets A and B are convex and compact. The functions fA and fB are continuous.


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    7. We are in a particular case of the general Nash Theorem [87],[3, Ch. These animals are in competition for a resource, for instance food. Inside this population, two behaviors coexist: The meetings between hawks and doves are characterized as follows. When two doves meet, they fairly share the resource R. This induces damages P and each hawk bring only half of the resource R minus the damages P. We assume that the resource R and the damages P are of the same kind. This game can be viewed as a two-player-game: