NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS
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Single Population Dynamics Under Migration Ritesh Agarwal and Somdatta Sinha Abstract—This paper presents analytical and numerical studies to show that discrete single population models can show distinctly different dynamics and survival-extinction behaviour under migration when compared to their behaviour under free growth. Keywords— Bellows Map, Migration, Discrete Growth, Bifurcations, Population Dynamics, Extinction, Chaos
I. I NTRODUCTION ODELLING population dynamics is an important area of study in Ecology, Epidemiology, Biodiversity and Conservation studies, as it allows one to predict the future trend and the effects of perturbation and environmental parameters on the population growth and size [1]. The size and dynamics exhibited by a population depend upon an number of factors. When resources are plenty, the individuals reproduce and the population size increases exponentially. But as the space occupied by the population gets crowded and resources are exhausted, the population moves towards its carrying capacity. This exerts a negative influence on reproductive output and survivorship, thereby acting to lower population growth rates. This density dependent feedback mechanism serves to regulate the population size in nature. The dynamics exhibited by the population depends on the relative strength of the density dependence and the reproductive output (growth rate). As the impact of density intensifies or reproductive capacity increases, the dynamics exhibited by a population which was displaced away from its carrying capacity will shift from a smooth approach to the carrying capacity. Which means shifting from monotonic damping, to more complex dynamics, such as damped oscillations, limit cycles, and eventually to the most extreme cases of density dependence, which exhibit a pattern known as chaotic dynamics. Ecological and demographic processes such as, migration, harvesting, culling, vaccination, quarantine, and segregation type of events also regulate population abundance. Dramatic examples include the precipitous drop of blue pike from annual catches of 10 million pounds to less than one thousand pounds in the mid 1950s, the unexpected collapse of the Peruvian anchovy population in 1973, and the sudden reduction of Great Britain’s grey partridge population in 1952. Some are due to excessive harvesting of fishes and some are attributed to gradual eutrophication of lakes. For many species, population growth is a seasonal affair. For example, many insects and plants have stages in life which do not overlap - the adult butterfly dies after laying eggs, or, the rice plant withers after the seeds mature. On the other hand, humans and many other organisms and plants exhibit continuous growth where the generations overlap. Therefore biological organisms exhibit two types of reproduction (termed as growth):(i) continuous, which is modelled using differential equations, and (ii)
M
Ritesh Agarwal is with the Department of Mathematics, IIT Kharagpur. Dr. Somdatta Sinha (
[email protected]) is with Centre for Cellular and Molecular Biology, Uppal Road, Hyderabad.
discrete, where the generations either do not overlap or there is age and stage structure of birth, which is modelled using difference equations. Discrete models for single populations are known to exhibit a variety of dynamics, from equilibrium to periodic, to period doubling bifurcations to chaos, with increasing growth rate [5]. But insect field population data seem to predominantly show equilibrium dynamics. Here we use a realistic single population model and show that migration, a common ecological process, can have quite counter-intuitive effects of on the population dynamics at different growth rates. A population with chaotic intrinsic growth rate can exhibit equilibrium dynamics under fixed migration. Thus ecological processes such as, constant migration, can affect the survival-extinction and population dynamics of single discretely growing organisms in nature. II. D ISCRETE S INGLE P OPULATION M ODEL The Bellows Model Here, we consider a discretely growing single population model which, in comparison to other density-dependent models, fits quite well to the insect population data set [2] [12]. This model, known as Bellows Map, is one of the commonly known one-dimensional discrete equations (maps) with two parameters, and is given by
Xt+1 = F (Xt ) =
RXt 1 + Xtb
(1)
where Xt+1 & Xt represent the population sizes at any two consecutive generations t and t + 1; F is a non-linear density dependent growth function which is a polynomial controlled by parameters with real values that incorporate all real world nonlinearity’s that may play a role in determining the size of the population in the next generation. R and b are the two parameters representing the intrinsic growth rate and intra-specific competition for resources. Increases in the parameter b intensifies the effects of population density. The density-dependence can be seen in the plot of Xt+1 versus Xt (the middle ”hump”-like curve in Figure( 3). The population grows exponentially when X is small, and then as X increases, the density dependence sets in and population size comes down. This leads to the single hump shape of the plot. The shape and peaked-ness of the hump is decided by both R and b. There are two kinds of intra-specific competition - ”Scramble” and ”Contest” - that regulate population density. Scramble involves equal partitioning of resources. In this case either all the individuals survive and reproduce, or, the entire population goes extinct when resources are insufficient. But in case of Contest, as the name suggests, some successful individuals get all they require, whereas some get insufficient resources for their survival and reproduction. These two are extreme forms, and
NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS
usually the situation is a mixture of the two. b = 1 for pure contest and b = ∞ for pure scramble. For 0 < b < 1 competition is only through different degrees of contest. All other values of b i.e. 1 < b < ∞, signify varying combination of scramble and contest in population growth. Here, we first study the survival-extinction behaviour and dynamics exhibited by single species Bellows Model, and then we study the effect of migration on it’s dynamics. We use steady state analysis and linear stability analysis to study the role of the parameters on population behaviour. We also perform numerical simulation to obtain the bifurcation diagrams for this model.
3.5
b=6 3
2.5
Population X
2
2
1.5
1
0.5
0
1
1.5
2
2.5
3 Growth rate R
3.5
4
4.5
5
Fig. 2. The bifurcation diagram at b = 6. 2
b=2
1.8
the population overshoots and undershoots the equilibrium state thereby giving rise to periodic variation in X. The bifurcations comes faster and faster with increasing R and b, ultimately turning the system chaotic, and the population visits infinitely many values [3].
1.6 1.4
Population X
1.2 1 0.8 0.6 0.4
5
0.2
4.5
0
1
1.5
2
2.5
3
3.5
4
4.5
4
5
Growth rate R
3.5 3 X(t+1)
Fig. 1. The bifurcation diagram at b = 2.
2.5 2
A. steady states and linear stability analysis
1.5 1
The steady states of equation( 1) are, X∗ = 0
and
X ∗ = (R − 1)1/b
0.5 0
0
0.5
1
1.5
2
2.5
3
3.5
4
X(t)
The stability conditions for the steady states are obtained as follows: For the steady state X ∗ = 0,
Fig. 3. Return Map for Bellows Model at R = 4, b = 6.
dF =R dX X ∗ =0
III. B ELLOWS M ODEL WITH CONSTANT MIGRATION
Thus, for 0 < R < 1 this steady state is stable and the population size always goes to zero. For R ≥ 1 this steady state is unstable. For the steady state X ∗ = (R − 1)1/b , b dF = − (b − 1) dX X ∗ =(R−1)1/b R Thus for 1 < R < b/(b − 2) this steady state is stable. Since the steady state X ∗ = 0 is unstable for R > 1, the population stabilizes to X ∗ = b/(b − 2) for 1 < R < b/(b − 2), beyond which the Bellows Map shows unstable dynamics. Figure( 1) shows the variation in the steady state for increasing intrinsic growth rate R for b = 2. The population dynamics is at equilibrium but the size increases with R. But for b > 2, the steady state becomes unstable and bifurcations set in. Figure( 2) shows the bifurcation diagram for b = 6, where the population size exhibits a variety of dynamics - from steady(equilibrium) state to chaotic through a hierarchy of stable cycles of period 2n (where n is a Natural Number) - with increasing growth rate R. As R increases, the equilibrium level of the population also rise. Then, as R rises further, density effect induces bifurcation and
The Bellows Model undergoing constant migration (L) at every generation can be written as: Xt+1 = F (Xt ) =
RXt +L 1 + Xt b
(2)
Where L is a parameter, which can take any real value. Figure ( 3) shows the return maps (Xt+1 versus Xt ) for the model with emigration (L < 0) and immigration (L > 0). The figure shows that under immigration the curve moves vertically up, and the identity line (F (Xt ) = Xt ) cuts the curve at just one point for Xt > 0. However under emigration, the curve moves vertically down, and the identity line intersects the curve at two points giving rise to two fixed points: x1 closer to origin, which is unstable, and x2 ). Depending upon the value of −L for a given R and b, there can also be a saddle node (equal fixed points), or no fixed point at all for Xt > 0. At R = Rmin the saddle node bifurcation occurs i.e. the map becomes tangential to the identity line (F (Xt ) = Xt ). For Bellows map equation( 1), R = 1 is Rmin , as the hump falls below the identity line for b < 1. Only for R > Rmin , there are two fixed points (one being the origin). So in order to have non trivial dynamical behavior, R must be greater than Rmin . If
ALIGARH MUSLIM UNIVERSITY, FEBRUARY 24-26, 2005
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Extiction 10 Emigtation rate (L)
R > Rmax , the system is in the escape state, i.e. F 2 (xc , R) < 0 and g n (Xt , R) → −∞. So what is required for a population to survive under emigration is that its intrinsic growth rate R should be Rmin < R < Rmax . This range depends on the strength of emigration. Bellows model without migration does not have a Rmax , because the tail of the hump extending to ∞. But under emigration (−L), when the hump moves down vertically, the hump can cross the X-axis at two points at Xt > 0. In this case Rmax exists.
Escape R
min
curve R
max
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Allee effect curve
A. Relationship between Rmin and L Here we calculate a relation between Rmin and L analytically to predict the minimum growth rate required for survival of a population undergoing emigration. The analysis is done only for b = 2, as values of b > 2 require complex computations. To find the relationship between Rmin and emigration L, we use the concept of saddle node bifurcation i.e. F 0 (Xt ) = 1 Taking x = Xt the equation( 2) reduces to, x4 + (R + 2)x2 − (R − 1) = 0 This equation has just one real positive root, r R + 2 1/2 R2 + 2R − } x={ 4 2
(3)
(4)
Now, x > 0 for every R > 1. If we express the relationship of Rmin and L explicitly using equation( 2) then, L=
Rmin x −x 1 + x2
(5)
The above equation gives the saddle node bifurcation curve (or, the line beyond which the population does not survive), indicating the maximum amount of emigration that a population can tolerate. The Rmin line is shown in the R − L parameter plot in figure( 4). B. Relationship between Rmax and L The relation between Rmax and L gives the demarcation line between survival and escape region when F 2 (xc ) ≥ 0, for b > 1. At b = 2 we have, F (F (x)) =
Rx R{ 1+x 2 − L}
Rx 2 1 + ( 1+x 2 − L)
−L
Let L = z + where,
R 3
R2 R2 )L − =0 4 2
z 3 + pz + q = 0
R2 R2 + (1 + R + ) p=− 3 4 2R3 R R2 R2 q=− + (1 + R + )− 27 3 4 2
P>=2 P=1 0
0
5
10
15 Growth rate (R)
20
25
30
Fig. 4. Stability and survival-extinction plot in (R − L), parameter space for b=2
Equation( 7) has three distinct positive roots for R > 14.8276, (see figure( 4)) because p3 q2 + 14.8275 can be seen by looking at the figure( 4). One will notice that a fast growing population goes extinct under a low emigration rate, but it survives under a range of higher emigration rate [7]. C. Bifurcation at b = 2 under emigration The relationship between R and b, when no migration takes place, is given by R = b/(b − 2). For b√< 2, the return map for L = 0 is globally stable because |F 0 ( R − 1)| < 1 for R > 1. However the same population under emigration(L < 0) can exhibit unstable dynamics through period doubling bifurcation. By using equation( 2) and solving F (x) = x we get, x3 − Lx2 − (R − 1)x − L = 0
(8)
Solving the above cubic equation, one gets the value of xp in terms of R, and substitute it in the next equation. The set of parameters (R, L) for which the fixed-point gives rise to an attractive cycle of order two (P = 2) is given by,
At x = (b − 1)−1/b = 1 we have, L3 − RL2 + (1 + R +
curve
(6)
(7)
F 0 (x) =
R(1 − x2 ) = −1 (1 + x2 )2
Let, y = x2 y 2 − (R − 2)y + (R + 1) = 0
Solution of the above quadratic equation gives, √ (R − 2) ± R2 − 8R y= 2
(9)
(10)
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NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS
Taking equation( 8) with the terms with L on the right hand side, and then squaring both sides we get, 2
2
2
2
2 2
x {x − (R − 1)} = L {1 + x }
If we take y = x2 and substitute the value of y from equation( 10) we have, y{y − (R − 1)}2 = L2 {1 + y}2
(11)
√
√ R2 − 8R ± R2 − 8R − R 2 √ { } = L2 2 R ± R2 − 8R Simplifying the above expression, √ (R2 − 4R − 8) ± R R2 − 8R 2 L = (12) 8 (R − 2) ±
Now the value of L in the equation( 12) is real iff R ≥ 8, So the minimum value of R where bifurcation begins is 8. Which is the tip of the P = 2 curve in figure( 4). If we take the limit of equation( 12) as R → − ∞ we will see that L = 0 is the asymptote to the bifurcation curve (see Appendix). D. Allee effect A population with low density is prone to extinction under emigration, or when low densities prevent individuals from finding mates [4]. This is called Allee effect in Ecology. In this model this occurs whenever F 2 (xc ) < x1 , where x1 is the repelling root of equation( 2) and xc is the critical point. Since x1 is deducted from F 2 (xc ), the region encompassed by the Allee effect is greater than the region bounded by the Rmax contour in figure( 4). Thus, for a specific value of R, the population succumbs to irreversible escape due to Allee effect at lower emigration rate than what is supported by Rmax contour. x1 is the repelling root and very close to the origin. So,
Considering (1 + x2 ) ≈ 1, we can write the value of x1 as: L R−1
Then equation f 2 (xc ) − x1 = 0 becomes, R{ R2 − L} L =0 −L− R 2 R−1 1 + ( 2 − L) Solving the above equation gives, L3 − RL2 + (R +
R2 R(R − 1) )L − =0 4 2
Our study with a realistic discrete population growth model undergoing migration shows interesting and non-intuitive results. The major result of the study are: a) The single population can show a variety of dynamics - from stable to periodic, period doubling and chaos - with changes in growth parameters ”R” and ”b”. b) The effect of constant migration on the population dynamics is growth-rate dependent. It shows the peculiar property of surviving at a higher emigration rate even though it goes extinct at a lower rate of emigration beyond a certain intrinsic growth rate of the population. c) A population that exhibits stable dynamics for given parameter values, can show periodic dynamics under emigration. Thus population dynamics under free growth and under emigration can be distinctly different, and hence, field populations may show different dynamics than what is expected of their growth parameters. Appendix To show that L = 0 is the asymptote to the bifurcation curve at the limit of equation( 12) as R → − ∞. √ (R2 − 4R − 8) − R R2 − 8R lim R→∞ 8 r 4 8 R2 8 {1 − − 2 − (1 − )} lim R→∞ 8 R R R q Now doing the taylors series expansion of (1 − R8 ) we get, 4 8 4 8 1 R2 {1 − − 2 − (1 − − 2 + O( 3 ))} R→∞ 8 R R R R R lim
Therefore, lim L = 0
R→∞
R EFERENCES
Rx −L=x 1 + x2
x1 =
IV. C ONCLUSIONS
(13)
This plot is marked as Allee effect curve in figure( 4). Thus we have presented a complete picture in figure( 4) of the dependence of the survival-extinction behaviour and population dynamics on the intrinsic growth rate of the organism and emigration rate.
[1] Charles J Krebs, Ecology, Harper & Row Publishers, 1972. [2] Hassell MP, Lawton JH and May, J Anim Ecol, Vol. 45, pp 471-486, 1994. [3] M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, Vol. 19, pp 459-467, 1978. [4] Sebastian J. Schreber,Chaos and Population Disappearances in Simple Ecological Models, Journal of Mathematical Biology, Vol. 42,pp 239-260, 2001. [5] R. M. May and G. F. Oster, Bifurcations and dynamics complexity in simple ecological models, Am. Nat., Vol. 110, pp. 573-599, 1976. [6] N. Parekh and Somdatta Sinha, Controlling dynamics in spatially extended systems, Phy. Rev. E, Vol. 65, pp. 0362271-9, 2002. [7] S. Sinha and S. Parthasarthy, Unusual dynamics of extinction in a simple ecological model, Proc. Nat. Acad. Sci 93,pp. 1504-1508, 1996. [8] J. D. Murray,Mathematical Biology, Biomathematics Text,pp 36-44,1989. [9] Somdatta Sinha,Are Ecological Systems Chaotic ? - An enquire into population growth models, Current science, Vol. 73(11), pp. 949-956,1997. [10] N. Parekh and Somdatta Sinha, Controllability of spatiotemporal systems using constant pinnings, Physica A, Vol. 318, pp. 200-212, 2003. [11] Somdatta Sinha and Parichay K. Das,Behaviour of Simple Onedimensional Maps Under Perturbation, Pramana-Journal of Physics, Vol. 48(1), pp. 87-98,1997. [12] T.S. Bellows, The descriptive properties of some models for density dependence, J. Anim. Ecol., Vol. 26, pp. 129-156, 1981.
