Control Schemes to Reduce Risk of Extinction in the Lotka-Volterra Predator-Prey Model

Transcription

1 Journal of Applied Mathematics and Phsics, 2014, 2, Published Online June 2014 in SciRes. Control Schemes to Reduce Risk of Etinction in the Lotka-Volterra Predator-Pre Model Jessica Li Kent Place School, Summit, USA Received 24 March 2014; revised 26 April 2014; accepted 3 Ma 2014 Copright 2014 b author and Scientific Research Publishing Inc. his work is licensed under the Creative Commons Attribution International License (CC BY). Abstract he Lotka-Volterra predator-pre model is widel used in man disciplines such as ecolog and economics. he model consists of a pair of first-order nonlinear differential equations. In this paper, we first analze the dnamics, equilibria and stead state oscillation contours of the differential equations and stud in particular a well-known problem of a high risk that the pre and/or predator ma end up with etinction. We then introduce eogenous control to reduce the risk of etinction. We propose two control schemes. he first scheme, referred as convergence guaranteed scheme, achieves ver fine granular control of the pre and predator populations, in terms of the final state and convergence dnamics, at the cost of sophisticated implementation. he second scheme, referred as on-off scheme, is ver eas to implement and drive the populations to stead state oscillation that is far from the risk of etinction. Finall we investigate the robustness of these two schemes against parameter mismatch and observe that the on-off scheme is much more robust. Hence, we conclude that while the convergence guaranteed scheme achieves theoreticall optimal performance, the on-off scheme is more attractive for practical applications. Kewords Lotka-Voterra, Predator-Pre Model, Etinction Control, Feedback Control, Stabilit 1. Introduction Predator-pre population dnamics are often modeled with a set of nonlinear differential equations. he Lotka- Volterra model [1] [2] is one of the simplest predator-pre models in which onl two species interact. Yet the model represents a powerful paradigm that can be etended to more sophisticated ecological dnamics such as How to cite this paper: Li, J. (2014) Control Schemes to Reduce Risk of Etinction in the Lotka-Volterra Predator-Pre Model. Journal of Applied Mathematics and Phsics, 2,

2 competition, disease and mutualism [3] and even economics [4]. Specificall, the Lotka-Volterra predator-pre model is a pair of first-order nonlinear differential equations = a b = c d where, represent the numbers of the pre and predator respectivel, and abcd,,, are positive parameters that describe the following dnamic interaction of the two species. Without the predator ( = 0), the pre growth rate is directl proportional to the population size. herefore the population will grow eponentiall. However, the predator reduces the pre growth rate, and the reduction is jointl proportional to, because the rate of predation upon the pre depends on the rate at which the two meet. Without the pre ( = 0), the predator growth rate is negative and directl proportional to the population size. herefore, the population will drop eponentiall. However, the pre increases the predator growth rate, and the increase is jointl proportional to, because the rate of predation upon the pre depends on the rate at which the two meet. he above Lotka-Volterra model (1) describes the autonomous dnamics between the two species without an eogenous input. In this paper, we introduce eogenous control to change the dnamics so as to achieve certain desired characteristics. In particular, it is well known [5] [6] that there is a high risk that the pre and/or predator ma end up with etinction in the Lotka-Volterra model. We introduce two control schemes to reduce the risk of etinction. he first control scheme guarantees that the dnamic sstem converges to a desired state according to an predefined trajector. he second control scheme uses a simple on-off logic to reduce the predator growth rate when needed. he on-off control scheme is eas to implement. Finall we stud the robustness of the two control schemes against parameter mismatch that often arises in practice. 2. Analsis of Lotka-Volterra Equation and Risk of Etinction Figure 1 illustrates an eample of the predator and pre population sizes varing with time. he population evolves in a periodic manner and there is a phase shift between the predator and pre populations. Initiall, both the pre and predator populations are small. Without enough food, the predator population shrinks and the pre population grows rapidl. After a dela, the remaining predator enjos sufficient food suppl and its population grows rapidl. his results in equall rapid drop in the pre population, which in turn leads to rapid drop in the predator population with a dela. he whole ccle repeats. We plot the predator and pre populations in the plane in Figure 2 and observe that the oscillate endlessl according to a fied contour. Along the contour, the following quantit C remains the same C = a ln + d ln b c (2) o see this, note that It follows that c d = = a b a b = c d a b c d = a d b d = c d aln b = c dln + C For comparison, Figure 2 plots the predator versus pre population contours of a variet of C resulting from different initial conditions. he equilibrium is obtained b solving (1) 645

3 Figure 1. Plot of predator and pre populations varing with a = d = 0.5, b= c = = 0 = 10. time ( ) ( ) Figure 2. Plot of predator versus pre population contours. a = d = 0.5, b = c = represents one of the two equilibrium d a points =, = c b. a b = 0 c d = 0 = =. he first equilibrium represents the scenario where both species becomes etinct. As shown in Figure 2, all of the contours oscillate around the second equilibrium. An important observation from both Figure 1 and Figure 2 is that in each ccle the predator and pre populations drop to a ver low level and then recover. In realit, other factors, such as disease and randomness, which have not been taken into account in the model, ma affect the population dnamics and cause the population to drop to zero. In one scenario, the pre ma become etinct first, which will then make the predator etinct too because of the lack of food source. Another possible scenario is that the predator ma become etinct first. With no predator, the pre population will grow to infinite. Of course in a real world other factors, e.g., food suppl for the pre, will eventuall limit the growth. While the particular dnamic pattern ma appear to be an artifact of the Lotka-Volterra model, the above ob- here are two solutions: { = = 0}, and { dc, ab} (3) 646

4 servation demonstrates certain phenomenon in some real world ecological environment where the predator-pre dnamics are so out-of-balance that the pre and/or predator end up etinction. In the following we will introduce a dnamic control mechanism to the Lotka-Volterra model so as to reduce the risk of etinction. 3. A Convergence Guaranteed Control Scheme Eogenous control mechanisms have been studied in the literature [7]-[9] to alter the dnamics of the pre and predator populations for a variet of control goals. he control goal of this paper is to reduce the risk of etinction. With eogenous control, the dnamic sstem is in general described as follows where (, ), (, ) population to a desired final state ( ), ( ) (, ) (, ) = a b u = c d u u u represent the control variables to be used. he first control scheme is to drive the t t, with constant,. Moreover, the trajector to- ward the final state is also to be controlled. As an eample, suppose that the control scheme is required to drive 0, 0, according to the first-order linear differ- from an initial population { ( ) ( )} to the final state { } ential equation d d ( ) dt ( ) dt ( ) = K ( ) = K Constant K, K specif how fast the population converges to the final state. hus, Constants, ( ) ( ) t = + ce t = + ce Kt Kt c c are determined b the initial condition { ( 0, ) ( 0) } he control scheme is thus given b It follows that ( 0) c = ( 0) c = (, ) ( ) a b u = K (, ) ( ) c d u = K (, ) = + ( ) (, ) = + ( ) u a b K u c d K Because the population is guaranteed to converge to the final state, this control scheme is called convergence guaranteed scheme. he control scheme is epressed in a closed-loop feedback form. From the closed form et, t, the trajector of the control variables is given b pressions of ( ) ( ) Kt Kt ( K+ K) t u( t) = ( a b ) + ( ac bc + K c) e bcxe bcce K K t t ( K+ K) t u ( t) = ( c d ) + ccye + ( ccx dc + K c) e + ccce Clearl the control is stable and converges to the stead state { a b, c d } gence rate depends on K, K. (4) (5) (6) (7) (8), and the conver- 647

5 Figure 3 shows the performance of the convergence guaranteed control scheme in order to drive the population to the desired final state. Figure 4 shows the control variables used to achieve the performance. We observe that with large K, K, the convergence becomes quicker to the final population state at the cost of more drastic control required. Note that in the convergence guaranteed control scheme the control variables u, u ma be negative. For eample, both u, u are initiall negative in Figure 4. In fact, the stead state itself is not necessaril positive for an abcd,,,,,. A negative control variable means that one has to add pre or predator to the population, which is much harder to implement in a real world than removal of the species. We net propose an on-off control scheme to address this problem. 4. An On-Off Control Scheme he above convergence guaranteed control scheme (7) achieves ver precise control of the population not onl the final state but also the trajector to the final state is precisel specified. In man real world scenarios, such a high precision ma not be required. In the following, our objective is to control the population to be within a desired dnamic range that is far from the risk of etinction. B relaing the control objective, we aim to greatl simplif the implementation of the control scheme. Specificall, consider the following on-off control scheme u (, ) ( ) u, = 0 (9) g, ϕ = (10) 0, otherwise Here, g is a positive constant and ϕ is a Boolean variable representing the control condition. he scheme applies onl to the predator. When condition ϕ is met, the eogenous control is to remove a faction g of the predator so as to help the pre population recover; otherwise, no control is eerted. he ke of the on-off control scheme is to design the control condition ϕ. We consider two design choices. he first choice is based on the population of the pre ϕ = 1, if < 0 (11) where 0 is a constant threshold. he idea is that the control is eerted if the pre population drops below threshold 0. Figure 5 shows the performance of this choice with 0 = 100. Apparentl, the pre population still drops well below 0, and more importantl, the predator population periodicall gets close to zero. One wa to view this on-off control scheme is to plot the predator versus pre population. Recall that with no control, the predator and pre populations of the original autonomous dnamic sstem (1) oscillate along a contour. If the control is constantl eerted, i.e., u (, ) = g, then the dnamic sstem would be the same as the original one ecept that d is replaced b d + g. When the control is turned on or off depending on the control condition ϕ, the predator and pre populations oscillate along a contour that switches between these two contours at = 0. hat is, on the left side of line = 0, the control is turned on and the contour follows the one u, = g, and on the right side the control is turned off and the contour follows the one with with ( ) (, ) 0 u =. Figure 6 plots the three contours. Clearl, with the on-off control scheme, the pre population is moved slightl awa from close to etinction. However, the predator is still near etinction periodicall. he second choice is based on the first-order derivative of the population of the pre ϕ = 1, if < α (12) where α is a constant threshold. he idea is that the control is eerted if the rate of decrease in the pre population drops below threshold α. Compared with (11), the second choice (12) ehibits a predictive capabilit in the sense that as long as the rate of change drops below the threshold, the control is turned on proactivel, even if the pre population itself ma not be ver low. We eamine the performance in Figure 7, which consists of four plots. he two plots in the left column correspond to α = 20 and initial condition ( 0) = ( 0) = 10. Comparing with the population dnamics with no control in Figure 1, the dnamic range of fluctuation is much narrower and the pre and predator are both far from the risk of etinction. It is interesting to note that intensive control is needed to drive the population from the initial condition to a desired oscillation; however, once the population reaches the desired oscillation 648

6 Figure 3. Performance of the convergence guaranteed control scheme. a = d = 0.5, b = c = (0) = (0) = 10. = 100, = 20. Figure 4. Control u, u in the scenario of Figure 3. Figure 5. Performance of the on-off control scheme with control condition (11) and 0 = 100, g =

7 Figure 6. Contour plot of Figure 5. Figure 7. Performance of the on-off control scheme with control condition (12) and g = 2. pattern, control is onl eerted ver occasionall. We make a similar observation for a different initial condition, e.g., ( 0) = 100, ( 0) = 2 as shown in the lower right plot. In the upper right plot, we change the threshold to α = 5. Compared with α = 20, control is now more read to be eerted. A comparison between these two thresholds of the upper two plots shows that with a smaller α, the dnamic range of the desired oscillation is narrower but it takes longer to drive from the initial condition to the desired oscillation. Figure 8 plots the predator versus pre population and shows the initial transition portion and the stead state oscillation contour. Comparison between Figure 8 and Figure 6 clearl shows that control condition (12) outperforms (11) to reduce the risk of etinction of both species. 5. Stud of Robustness In this section, we investigate the robustness of these two schemes against parameter mismatch. Specificall, so far we have assumed that the control scheme knows eactl the model parameters ( abcd,,, ). In practice, however, the precise values of the parameters ma not be available. Usuall people have to monitor the pre and predator population dnamics to estimate the parameters. More importantl, the parameters ma change over time, therefore causing parameter mismatch. o stud the robustness, we assume that the actual parameters are slightl different from those used in the control scheme. hat is, the control scheme assumes the following set of parameters: aˆ = dˆ = 0.5, bˆ = cˆ =

8 Figure 8. Plot of predator versus pre population of the case of α = 20, (0) = (0) = 10 in Figure 7. Figure 9. Robustness against parameter mismatch. he parameters of the actual model are given b a = 0.6, d = 0.4, b = 0.012, c = Figure 9 compares the robustness performance of the two control schemes. he upper two plots are for the convergence guaranteed scheme, and the lower two are for the on-off scheme. In both cases, the left one is without parameter mismatch and the right one is with parameter mismatch. We observe that the on-off scheme is much more robust than the convergence guaranteed scheme, as the population dnamics hardl change. In fact, the convergence guaranteed scheme becomes instable. 6. Conclusion In this paper, we have proposed two control schemes to reduce the risk of etinction in the Lotka-Volterra predator-pre model. he convergence guaranteed scheme achieves ver fine granular control of the pre and predator populations, in terms of the final state and convergence dnamics, at the cost of sophisticated implementation. he on-off scheme is ver eas to implement and drive the populations to stead state oscillation that is far from the risk of etinction. We furthermore investigate the robustness of these two schemes against parameter mismatch and observe that the on-off scheme is much more robust. Hence, we conclude that while the convergence guaranteed scheme achieves theoreticall optimal performance, the on-off scheme is more attractive for practical applications. A net step of the research would be to appl the on-off scheme to other predator-pre models and investigate the robustness not onl against parameter mismatch but also against model mismatch. 651

9 References [1] Lotka, A.J. (1910) Contribution to the heor of Periodic Reaction. he Journal of Phsical Chemistr, 14, [2] Volterra, V. (1931) Variations and Fluctuations of the Number of Individuals in Animal Species Living ogether. In: Chapman, R.N., Ed., Animal Ecolog, McGraw-Hill, New York, [3] Holt, R.D. and Pickering, J. (1985) Infectious Disease and Species Coeistence: A Model of Lotka-Volterra Form. he American Naturalist, 126, [4] akeuchi, Y. (1996) Global Dnamical Properties of Lotka-Volterra Sstems. World Scientific, Singapore. [5] Mollison, D. (1991) Dependence of Epidemic and Population Velocities on Basic Parameters. Mathematical Biosciences, 107, [6] Wikipedia (2014) Lotka-Volterra Equation. [7] Leung, A.W. (1995) Optimal Harvesting-Coefficient Control of Stead-State Pre-Predator Diffusive Volterra-Lotka Sstems. Applied Mathematics & Optimization, 32, [8] El-Gohar, A. and Yassen, M.. (2001) Optimal Control and Snchronization of Lotka-Volterra Model. Chaos, Solitons & Fractals, 12, [9] Chen, F. (2005) Positive Periodic Solutions of Neutral Lotka-Volterra Sstem with Feedback Control. Applied Mathematics and Computation, 162,

10 Scientific Research Publishing (SCIRP) is one of the largest Open Access journal publishers. It is currentl publishing more than 200 open access, online, peer-reviewed journals covering a wide range of academic disciplines. SCIRP serves the worldwide academic communities and contributes to the progress and application of science with its publication. Other selected journals from SCIRP are listed as below. Submit our manuscript to us via either submit@scirp.org or Online Submission Portal.