Global Dynamics of Some Periodically Forced, Monotone Di erence Equations

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1 Global Dynamics of Some Periodically Forced, Monotone Dierence Equations J. M. CUSHING Department of Mathematics University of Arizona Tucson, AZ SHANDELLE M. HENSON Department of Mathematics The College of William and Mary Williamsburg, VA J. Dierence Equations and Applications 7 (00), Dedicated to C. D. Ahlbrandt on the occasion of his sixtieth birthday Abstract We study a class of periodically forced, monotone dierence equations motivated by applications from population dynamics. We give conditions under which there exists a globally attracting cycle and conditions under which the attracting cycle is attenuant. Keywords: dierence equations, periodic forcing, global attractors, periodic cycles, cycle average AMS Classication codes: 39A, 9D5 Introduction If () is an increasing function of the solution sequences of the dierence equation + = ( ) =0 () are monotonic and hence, if bounded, converge. Assuming () is continuous and letting in () we nd that the limit =lim of a convergent solution is necessarily an equilibrium solution,since is a root of the equilibrium equation = () We are motivated by dierence equations of this monotone type that appear as applications in population dynamics. An example is the Beverton-Holt equation + = 0 0 +

2 where and are positive constants. If the solution of this initial value problem converges to the equilibrium =0. If the solution converges to the positive equilibrium = =. With introduced explicitly, the Beverton-Holt equation becomes + = +( ) Because all solutions are monotonic and converge, this dierence equation is, from a dynamics point of view, a more appropriate analog to the famous logistic dierential equation than are non-monotonemapssuchastheso-called discrete logistic equation + = ( ) (with = )ortherickerequation + = (with = ln ). As is well known, such one hump maps can have periodic solutions and even chaotic solutions. In theoretical ecology, the parameters and play an important role. The coecient is considered a characteristic of the population (its inherent growth rate ), determined by life cycle and demographic properties such as birth rates, survivorship rates, etc. The coecient is considered a characteristic of the habitat or environment (called the carrying capacity ), e.g., resource availability, temperature, humidity, etc. The autonomous equations and their equilibrium theory above are appropriate in biological applications only if the and are constant over time. In cases where one or both of these parameters uctuate in time (which is, of course, quite common for biological populations), the model equations become non-autonomous. For example, periodic uctuations are common (caused, for example, by annual or daily uctuations in the physical environment), in which case and/or become periodic functions of time. While periodic dierential equation population models have been considered in the literature, relatively little attention has been paid to periodic dierence equation population models. (See, however, [3], [5], [6], [7], [8], [9].) One ecological question that has been studied by means of periodic models concerns the eect of a periodic environment on a population, i.e., the eect of aperiodic. Attentionhasfocusedonwhetherornotapopulationisadversely aected by a periodic environment (relative to a constant environment of the same average carrying capacity). Early results based on the logistic dierential equation implied that a periodic carrying capacity is deleterious in the sense that the average of the resulting population oscillations is less than the average of [], [], [0], []. Later results showed this assertion can be model dependent [4], []. A recent study utilizing non-monotone dierence equations has demonstrated the latter point. In [3], [9] it is shown, by mathematical analysis and laboratory experiments, that it is possible for a periodic environment to be advantageous for a population in the sense that average densities are greater in aperiodicenvironmentthaninaconstantenvironment. In this paper, we show that aperiodicenvironmentisalwaysdeleteriousfor populations modeled by a class of monotone dierence equations. This result is analogous to that in [], [0] for the logistic dierential equation. In a periodic environment version of the Beverton-Holt equation we replace by a periodic sequence = (). Wewillrestrictourattentioninthispaper

3 to oscillations of period two. Therefore, we take = ³ + () where is the average of over time and [0 ) the relative amplitude of the oscillation. (We have arbitrarily chosen the oscillation so that its maximum occurs at even time units. Our results remain valid if instead the maximum occurs at odd time units, i.e., ( 0].) In the resulting equation, we can eliminate the parameter by the rescaling = and obtain the equation + = +( ) () +() Motivated by the periodically forced Beverton-Holt equation () we consider a general class of periodically forced equations of the form µ + = + () [0 ) (3) where the function satises the conditions () $ () A: 0 ( + + ) ( ) 0 () 0, 00 () 0 for all + 0 (0) = 0 () Here + =[0 ), + 0 =(0 ) and () =lim (). Dene $ 0 (0+). By A, this right hand limit exists in the extended sense, i.e., 0. Consider equation (3) when =0 i.e., consider the unforced, autonomous equation + = ( ) (4) If assumption A implies () for 0. In this case, solutions converge to the equilibrium 0 for all positive initial conditions 0 0 In this case, we say 0 is globally attracting for 0 0. On the other hand, if then there exists a unique positive root 0 of the equation = (). Inthis case, solutions converge to the equilibrium for all positive initial conditions 0 0, andwesay is globally attracting for 0 0. Lemma Assume A. If then the equilibrium 0 of (4) is globally attracting for 0 0. If (including = ) equation(4)hasaglobally attracting positive equilibrium. By a two-cycle we mean a solution of (3) whose even and odd iterates are both constant, i.e., = 0 and + = for all =0. We will denote a two-cycle by the ordered pair ( 0 ).Ifboth 0 0 and 0 then 3

4 we say the two-cycle is positive. Thesolutionof (3) converges to a two-cycle ( 0 ) if lim ( + )=( 0 ). (5) Atwo-cycle( 0 ) is globally attracting for 0 0 if solutions of (3) converge to ( 0 ) for all initial conditions 0 0. In Section we consider the asymptotic dynamics of the periodically forced equation (3). Theorem establishes, when, theexistenceofa(unique) positive two-cycle that is globally attracting for 0 0. Ourmainresultappears in Section 3. Theorem 3 shows, under certain concavity conditions, that the globally attracting two-cycle ( 0 ) when is attenuant, i.e., it has a suppressed average in the sense that ( 0 + ). This means the presence of periodic forcing in equation (3), i.e., 0, results in a global attractor with decreased average. We will have need of the following two assumptions. Note A and A3 together imply A. Two-cycle Solutions A: 0 ( + + ) ( ) 0 () 0 for all + 0 A3: 0 ( + + ) ( ) 00 () 0 for all + 0 A implies the solution of the periodically forced equation (3) with 0 0 is positive, i.e., 0 for all =0.Ourgoalinthissectionisdetermine when it is true that for all 0 0 the solution tends to a unique positive twocycle solution. The components of a two-cycle ( 0 ) necessarily satisfy the pair of two-cycle equations ³ = 0 ³ 0 = + 0 Conversely, positive solutions 0 0, 0 of these simultaneous equations dene a positive two-cycle ( 0 ). Theorem Assume A and consider the periodically forced equation (3) for 0. If then the equilibrium 0 is globally attracting for 0 0. If (including = ) thenthereexistsapositivetwo-cycle( 0 ) that is globally attracting for 0 0. If() is increasing (e.g., if A holds), then this two-cycle is strict, that is to say 0 6=.. (6) 4

5 Proof. Assume 0 0. From the rst composite of equation (3) we see that the even and odd iterates, and + of the solution with initial condition 0 0 satisfy the equations + = ( ) 0 = 0 0 (7) + = ( ) = 0 respectively, where µ µ () $ + µ µ () $ + µ + µ Here = and = + for =0.Alternativelywecanwrite µ µ + () = ( ) + µ µ () = (+) + Assumption A implies () and () also satisfy A. A calculation shows 0 (0+) = 0 (0+) = From Lemma applied to both equations in (7) we conclude that both sequences, and hence converge to 0 if. If, on the other hand, then each equation in (7) has a globally attracting, positive equilibrium. Thus, the even iterates = converge to the unique positive root 0 0 of = () and the odd iterates = + converge to the unique positive root 0 of = (). If we let = in equation (3) and let,thenwend by continuity that 0 and satisfy the rst of the two-cycle equations (6). Similarly, letting = + and passing in (3) we nd that 0 and also satisfy the second equation in (6). Thus, the two constants 0 and dene a positive two-cycle ( 0 ) to which converges. If 0 = = then from the two-cycle equations (6) it follows that µ + = µ If is increasing, we arrive at a contradiction. Thus, in this case, 0 6= and the two-cycle is strict. 3 Attenuant two-cycles For (including = ) theautonomousequation(4)hasauniquepositive equilibrium 0. Changing variables from to in equation (4), we 5

6 can assume without loss in generality that the unique positive equilibrium is. Under this assumption () satises the added restriction A4: () = (and hence () = ). Note that A and A4 together imply. Thus,under these two assumptions Theorem implies the existence of a positive two-cycle ( 0 ) that is globally attracting for 0 0. Thistwo-cycleisattenuantifand only if ( 0 + ) (8) Theorem 3 Assume A, A3 and A4. For each (0 ) the positive, globally attracting (for 0 0), strict two-cycle of the periodically forced equation (3) is attenuant Before proving this theorem we establish some lemmas. Lemma 4 Assume A and A. For let ( 0 ) be the positive two-cycle guaranteed by Theorem. Then 0 0 for each (0 ). Proof. For purposes of contradiction assume 0. From A and the second of the two-cycle equations (6) we have µ µ µ 0 = =(+) + + By A and the rst of the two-cycle equations (6) we have µ µ 0 ( + ) ( + ) + + µ 0 = 0 = + which gives the contradiction 0. Lemma 5 Assume A and A4. Then µ ( + ) + for all (0 ) +( ) Proof. For [0 ) dene the function () $ µ ( + ) + +( ) µ µ 6

7 Since (0) = it suces to show () is a decreasing function of For (0 ) acalculationshows where 0 () =( + ) ( ) () $ µ µ 0 By the Mean Value Theorem, 0 () = 0 () for some ( +) However, by A 0 () = µ and hence 0 () 0. Lemma 6 Assume A, A3 and A4. Then 0 ³ + for all [0 ) Proof. Since 00 () 0 for all 0 it suces to show 0 By the Mean Value Theorem, = () = 0 () for some However, A3 implies 0 () 0 and thus 0 + = 0 Proof of Theorem 3. If the two-cycle is attenuant by Lemma 4. Therefore we assume. Wecanrewritethetwo-cycleequations(6)as µ 0 = (+) + µ 0 = ( ) Consider the second order Taylor expansions at = µ µ µ ( ± ) =(± ) + 0 ( ) + ± ( ) ± ± ± where ± ( ) $ µ 00 ± ± ( ) and ± is a nonzero number between and By A, ± ( ) 0 The twocycle equations become µ µ (a) = (+) + 0 ( 0 ) + + ( 0 ) + + (9) µ µ (b) 0 = ( ) + 0 ( ) + ( ). 7

8 Note µ µ 0 ( ) 0 ( ) + since 00 () 0 for 0. Thus,from(9b)wehave µ µ 0 ( ) + 0 ( ) + If we add (9a) to both sides of this inequality, we obtain ( 0 + ) µ µ ( + ) +() + µ µ ( 0 + ) Lemma 5 implies ( 0 + ) + 0 µ µ + ( 0 + ) and hence µ µ µ ( 0 + ) By Lemma 6 the second factor is positive and therefore the rst factor is negative. By inequality (8), it is certainly true that a two-cycle ( 0 ) is attenuant if max { 0 } In this case, we call the two-cycle strongly attenuant. If, on the other hand, an attenuant two-cycle satises max { 0 } we call the two-cycle weakly attenuant. Theorem 7 Assume A, A3 and A4. For (0 ) suciently close to 0 the positive, globally attracting (for 0 0), strict two-cycle ( 0 ) of the periodically forced equation (3) is weakly attenuant For (0 ) suciently close to the two-cycle is strongly attenuant Proof. From the two-cycle equations (6) we have the equation µ µ µ = + (0) and hence = µ µ µ + for 0. ByAandA,therighthandside,whichisidenticalto µ µ µ +, () 8

9 is a decreasing function of 0 Therefore, any positive root of () is unique. In particular the root =when =0unique. The implicit function theorem implies there exists a (locally unique), continuously dierentiable root = () for near 0 satisfying (0) =. Dierentiating both sides of equation () with respect to and evaluating the result at =0we obtain 0 (0) = 0 () + 0 (). From 0 () 0 and 0 0 () = + 0 () it follows that 0 (0) 0. Thus, () for 0 suciently small and the two-cycle is weakly attenuant. From equation (0) we derive the inequalities µ µ 0 =(+) + µ µ = (+) + µ ( + ) () + valid for all (0 ). Thisimplies lim () =0 µ + which implies for near By Lemma 4, max { 0 } and the two-cycle is strongly attenuant for near. 4 Examples Consider the periodically forced Beverton-Holt equation (). For this equation () = +( ) () = +( ), All assumptions A, A3 and A4 (and hence A) are satised. Therefore, for each (0 ) the equation () has an attenuant, positive, strict two-cycle ( 0 ) that is globally attracting for 0 0. In fact, in this example the two-cycle is ( 0 ) where and 0 = ( +) + ( + ) = ( 0 + )= ( +) + + ( ) + + ³ The cycle is weakly attenuant ( ) for0 +. SeeFigureforanexample. for + 9 and strongly attenuant

10 . c (c 0 + c )/ 0.6 c FIGURE. The two-cycles of the periodically forced Beverton-Holt equation () and their averages are plotted as functions of the relative amplitude when =5. and Other examples to which Theorems, 3 and 7 apply are () = for 0, +( ) () = for 0. The latter example is a case when =. Figure illustrates this example when =...0 c 0.8 (c 0 + c )/ 0.6 c FIGURE. The two-cycles of the periodically forced equation (3), with () =,andtheiraveragesareplottedasfunctionsofthe relative amplitude. Inthiscase =. 0

11 References [] B. D. Coleman, On the growth of populations with narrow spread in reproductive age. I. General theory and examples, J. Math. Biol. 6(978), -9 [] C. S. Coleman and J. C. Frauenthal, Satiable egg eating predators, Math. Biosci. 63 (983), 99-9 [3] R. F. Costantino, J. M. Cushing, B. Dennis, R. A. Desharnais, S. M. Henson, Resonant population cycles in temporally uctuating habitats, Bull. Math. Biology 60, No. (998), [4] J. M. Cushing, Oscillatory population growth in periodic environments, Theor. Pop. Biol. 30 (987), [5] J. M. Cushing, Periodically forced nonlinear systems of dierence equations, J. Di. Eq.Appl.3(998), [6] S. M. Henson, Existence and stability of nontrivial periodic solutions of periodically forced discrete dynamical systems. J. Di Eq. Appl. (996), [7] S. M. Henson, The eect of periodicity in maps, J. Di Eq. Appl. 5(999), 3-56 [8] S. M. Henson, Multiple attractors and resonanceinperiodicallyforcedpopulation models, Physica D 40 (000), [9] S. M. Henson and J. M. Cushing, The eect of periodic habitat uctuations on a nonlinear insect population model, J. Math. Biol. 36 (997), 0-6 [0] R. M. May, Stability and complexity in model ecosystems, PrincetonUniversity Press, Princeton, New Jersey, 973 [] R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations, Wiley & Sons, New York, 98 [] S. Rosenblat, Population models in a periodically uctuating environment, J. Math. Biol. 9(980),3-36

Signature Function for Predicting Resonant and Attenuant Population 2-cycles

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