MATHEMATICAL BIOSCIENCES AND ENGINEERING Volume 4, Number 4, June 2007
http://math.asu.edu/˜mbe/ pp. 1–11
SOME MECHANISTICALLY DERIVED POPULATION MODELS
Yang Kuang Department of Mathematics and Statistics Arizona State University, Tempe, AZ 85287-1804
(Lecture note for June 13, 2007 at MTBI)
Abstract. The purpose of this talk is to mechanistically formulate a series of mathematically tractable population growth models that are implicitly or explicitly dependent on resource dynamics.This series of models include Logistic equation, Holling type II predator prey model, Beddington-DeAngelis model, ratio-dependent model and the more recent stoichiometric population growth models.
1. Introduction. Predator-prey models are arguably the most basic and important building blocks of the bio- and ecosystems as biomasses are grown out of their prey/resource masses. Species compete, evolve and disperse simply for the purpose of seeking key resources to sustain their live-long struggle for their very existence. Depending on their specific settings of applications, they can take the forms of single species growth model where resource dynamics is assumed to be at a much faster pace, resource-consumer, plant-herbivore, parasite-host, tumor cells (virus)immune system, susceptible-infectious interactions, etc. They deal with the general loss-win interactions and hence may have many applications outside of ecosystems. When seemingly competitive interactions are carefully examined, they are often in fact some forms of predator-prey interaction in disguise. Most of existing population growth models are highly ad hoc and phenomenological at best. Most are formulated to serve theoretical considerations of some general hypotheses. These simple looking mathematical models, while highly tractable, are often disconnected to specific lab or field observations. Indeed, the lack of biological mechanisms can often lead misleading or even erroneous biological findings. For example, it is often regarded that the growth rate (birth rate - death rate)and carrying capacity in the logistic equation are independent parameters. In fact, some people even regard the growth rate as death rate. The so-called paradox of enrichment (Rosenzweig, M.L. 1971) is also proven to be more of a model artifact than reality. In general, careless model formulation and the even a misusing of the terminology can generate erroneous statements. For example, the use of mass action transmission term in the classical Kermack-McKendrick model lead many to misinterpret a simple local stability result to imply that there is a minimum threshold of initial susceptible population level in order for an epidemic to take hold (p246, 2000 Mathematics Subject Classification. 92D25 and 34C60. Key words and phrases. Droop model, logistic equation, predator-prey model, functional response, stoichiometry.
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Edelstein-Keshet 2005). The Kermack-McKendrick model consists of a system of three coupled nonlinear ordinary differential equations, S 0 = −βd SI, I 0 = βd SI − νI, R0 = νI.
(1.1)
Here βd S is the density dependent infection rate (may lack of any real biological basis), and ν is the recovery rate. The key value governing the time evolution of these equations is an epidemiological threshold that often referred as basic reproduction (reproductive) ratio, or basic reproduction (reproductive) number R0 = βd S(0)/ν.
(1.2)
R0 is defined as the number of secondary infections caused by a single primary infection. It determines the number of people infected by contact with a single infected person before his death or recovery. Observe that the total population N = S+I+R is constant. Hence the Kermack-McKendrick model is mathematically equivalent to S 0 = −βSI/N, I 0 = βSI/N − νI, R0 = νI. (1.3) However, here β is to be understood as the biologically more plausible maximum (density independent) transmission rate. For this model, R0 is independent on the initial susceptible population level R0 = β/ν.
(1.4)
If mathematical biosciences models are to be truly useful and to possess predictive power like many physical and engineering models, they shall be carefully formulated to incorporate basic yet crucial biological (such as resource driven dynamics, stoichiometric constraints) mechanisms and confirm to basic mathematical (logical consistence) and physical (conservation laws) principles. This talk wants to show you via many familiar model examples that this ambitious goal is achievable. 2. A mechanistic derivation of the logistic equation. Population growth involves and often is determined by the birth and death processes. Most of the existing studies focus on birth process on the combined birth and death processes (the growth process). In general, death mechanisms are more numerous and difficult to study then birth mechanisms in a lab or field setting. In a short time frame, growth dynamics can be approximated by a linear differential equation with the coefficient called growth rate. Longer term, this growth rate shall be regarded as time dependent, or density dependent. The so-called Droop equation provides a time and experiment tested simple mathematical expression for biomass growth rate. We shall show that it also provides a convenient base for deriving the classical logistic equation. This section is adapted from Kuang et al. (Kuang et al. MBE, 2004). In 1968, Droop reported some surprising findings based on his most ambitious and comprehensive chemostat experiment to date in terms of concept, technical difficulty and mathematical analysis, it was to surpass by far all that had gone before (Leadbeater, 2006). The experiment studied the kinetics of vitamin B12 limitation in Monochrysis lutheri in continuous and exponentially growing batch cultures and in washed cell suspensions. The aim of this experiments was to relate specific growth rate to substrate concentration (Droop 1966b). Contrary to conventional belief, the specific growth rate (m) of Monochrysis in the chemostats was found not to depend directly on medium substrate concentration. However, the one relationship that did stand out was that growth depended on the intracellular concentration of vitamin
SOME MECHANISTICALLY DERIVED POPULATION MODELS
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Plot of Droop function 0.9 0.8 0.7
µ (Q)
0.6 0.5
q=3, µm=0.75
0.4 0.3 0.2 0.1 0
0
10
20
30
40 Q
50
60
µ Figure 1. Plot of Droop function µm 1 −
70
q Q
80
¶ .
B12 (cell quota Q). The relationship between specific growth rate (µ ) and cell quota (Q) took the following simple form (Droop 1973, 1974). µ ¶ q µ = µm 1 − . (2.1) Q µ ¶ q This equation is called the Droop equation. We call µm 1− Q the Droop function. The parameter q is the minimum quota necessary for life (the subsistence quota) and represents the value of cell quote Q at zero growth rate. µm is the growth rate at infinite internal nutrient content which is clearly unattainable in experiments. Our main purpose in this section is to derive the logistic equation via Droop equation. We consider a single species growing in a closed environment where there is a single limiting nutrient. For convenience, we assume below this limiting nutrient is phosphorous P . Hence the total amount of phosphorus Pt in the environment remains constant. If we let Px and Pf be the phosphorus in the species and the free phosphorus respectively, then Pt = Px + Pf . Let x = x(t) be the species density and Q = Q(t) be the species’ cell quota for P . Then Pp = Qx. Hence Pt = Pf + Qx.
(2.2)
In the following, we let q be the species’ minimal cell quota for P , µm be the species’ maximal growth rate, D be its death rate. By (2.1), we have the following equation for the species growth: µ ¶ q dx = µm 1 − x − Dx. (2.3) dt Q We need an equation governing the dynamics of Q, the species’ cell quota for P . We assume that Q’s recruitment comes proportionally from the free phosphorus (αPf ) and its depletion because of cell growth is µm (Q − q). This results in the
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following simple equation dQ = αPf − µm (Q − q). (2.4) dt We assume that Q(0) ≥ q. Mathematically, this ensures that Q(t) ≥ q for all t > 0. Since the cell metabolic process operates in a much faster pace than the growth of total biomass of a species, the quasi-steady-state argument allows us to approximate Q(t) by the solution of αPf − µm (Q − q) = 0, (2.5) which takes the form of Q=
αPf + qµm . µm
(2.6)
This together with (2.2) yields
µ ¶ µm Pf = Pt − qx . µm + αx
Substituting (2.7) into (2.6) yields Q=q+
µ ¶ α Pt − qx . µm + αx
(2.7)
(2.8)
Substituting the above into (2.3) and applying some straightforward simplification yields dx Pt − qx = µm x − Dx. (2.9) dt Pt + µm qα−1 The above equation can be rewritten as µ ¶ dx qx + µm qα−1 = µm x 1 − − Dx. (2.10) dt Pt + µm qα−1 We can rewrite the above equation as · ¸ x + µm α−1 dx = (µm − D)x 1 − . dt [(µm − D)/µm ][µm α−1 + Pt /q]
(2.11)
Or equivalently,
· ¸ dx (µm − D)Pt /q − Dµm α−1 x = x 1 − . (2.12) dt Pt /q + µm α−1 [(µm − D)Pt /(qµm ) − Dµm α−1
It clearly takes the form of the classical logistic equation · ¸ dx x = rx 1 − dt K with r=
(2.13)
(µm − D)Pt /q − Dµm α−1 Pt /q + µm α−1
and K = [(µm − D)Pt /(qµm ) − Dµm α−1 . It is easy to observe that both r and K are increasing functions of α and decreasing dunction of D. This makes good biological senses. It should be pointed out here that we did not assume the population suffers from a crowding effect explicitly. However, this crowding effect is implicitly provided by the fact that the total nutrient in the system (here P ) is fixed, and individuals have to compete for this
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resource. Observe that instead of the often-assumed carrying capacity of the form Pt /q, here the carrying capacity has the expression of K = [(µm − D)Pt /(qµm ) − Dµm α−1 .
(2.14)
This says that although theoretically the environment may accommodate Pt /q plants, the actual upper limit the plant biomass can attain is K = [(µm −D)Pt /(qµm )− Dµm α−1 , less than that. The reason the maximal carrying capacity Pt /q cannot be reached in practice is that the death toll in a population keeps the population below its potential maximum. It says clearly that a population with a relatively low death rate will likely amass more biomass than a population with a relatively high death rate. 3. Holling type predator-prey models. The previous section shows that logistic growth model is a predator-prey model in disguise. However, in logistic growth model, the total resource amount is constant, even though the amount of free nutrient is dynamic. In addition, it is assumed that the nutrient can neither be produced, nor destroyed. In most models commonly labeled as predator-prey models, the preys or resources are usually allowed to grow and die in highly dynamical ways. The simplistic examples of such ways for growth dynamics include exponential growth for the classical Lotka-Volterra predator-prey model, dx dy = ax − bxy, = cxy − dy, (3.1) dt dt the logistic growth for Holling type II predator-prey model (also called RosenzweigMacArthur predator-prey model) ³ dx x´ bxy dy cxy = rx 1 − − , = − dy. (3.2) dt K a+x dt a+x The death dynamics is usually dominated by the predation and such terms are frequently expressed in the form of yp(x) and f (x) is called the predator’s functional response function, or simply, functional response. In the Lotka-Volterra predatorbx prey model, p(x) = bx while in Holling type II predator-prey model p(x) = a+x . This begs the question of how to mechanistically derive f (x). The so-called Gausetype predator-prey model is a more general and also classical form for predator-prey models (Gause 1934, Freedman 1980) dx dy = xg(x) − yp(x), = yq(x) − dy. (3.3) dt dt In such form, q(x) is referred as the predator’s numerical response function, or simply, numerical response. In most of the existing model, q(x) = cp(x) where c is the conversion or yield constant. Both notions of functional and numerical responses are coined by Holling (1959), who studied predation of small mammals on pine sawflies. He found that predation rates increased with increasing prey population density. This resulted from two distinct effects: (1) each predator increased its consumption rate when its prey density is increased, and (2) predator density increases after prey density is increased. Holling considered these effects as two kinds of responses of predator population to prey density: (1) the functional response and (2) the numerical response. Holling (1959) suggested a model of functional response which is often called “disc equation” because Holling used paper discs to simulate the area examined
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by predators. It is equivalent to the model of enzyme kinetics developed by Lenor Michaelis and Maude Menten in 1913 (Sharov 1996). This model illustrates the principal of time budget in behavioral ecology. It assumes that a predator spends its time on two kinds of activities: 1. Searching for prey 2. Prey handling which includes: chasing, killing, eating and digesting. Consumption rate of a predator is limited in this model because even if prey are so abundant that no time is needed for search, a predator still needs to spend time on prey handling. Total time T equals to the sum of time spent on searching Ts and time spent on handling Th . Assume that a predator captured Ha prey during time T . Handling time should be proportional to the number of prey captured Th = Ha h where h is time spent on handling of one prey. We also assume that a predator searches area a (search rate) per unit of time and catch a fixed proportion ρ of all prey in there. Let x be the prey density, then Ha = aρxTs . Hence Ts =
Ha . aρx
Therefore T = Ts + Th = and hence
Ha + Ha h aρx
aρxT . 1 + aρhx This gives the functional response of the form of aρx p(x) = . 1 + aρhx Clearly, at low prey densities, predators spend most of their time on search, whereas at high prey densities, predators spend most of their time on prey handling. Holling (1959) considered three major types of functional response: Type I functional response is found in passive predators like spiders. The number of flies caught in the net is proportional to fly density. Prey mortality due to predation is constant (right graph on the previous page). Type II functional response is most typical and corresponds to the equation above. Search rate is constant. Plateau represents predator saturation. Prey mortality declines with prey density. Predators of this type cause maximum mortality at low prey density. For example, small mammals destroy most of gypsy moth pupae in sparse populations of gypsy moth. However in high-density defoliating populations, small mammals kill a negligible proportion of pupae. Type III functional response occurs in predators which increase their search activity with increasing prey density. For example, many predators respond to kairomones (chemicals emitted by prey) and increase their activity. Polyphagous vertebrate predators (e.g., birds) can switch to the most abundant prey species by learning to recognize it visually. Mortality first increases with prey increasing density, and then declines. If predator density is constant (e.g., birds, small mammals) then they can regulate prey density only if they have a type III functional response Ha =
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because this is the only type of functional response for which prey mortality can increase with increasing prey density. However, regulating effect of predators is limited to the interval of prey density where mortality increases. If prey density exceeds the upper limit of this interval, then mortality due to predation starts declining, and predation will cause a positive feed-back. As a result, the number of prey will get out of control. They will grow in numbers until some other factors (diseases of food shortage) will stop their reproduction. This phenomenon is known as “escape from natural enemies” discovered first by Takahashi. Numerical Response means that predators become more abundant as prey density increases. Reproduction rate of predators naturally depends on their predation rate. The more prey consumed, the more energy the predator can allocate for reproduction. Mortality rate also reduces with increased prey consumption. The simplest model of predator’s numerical response is based on the assumption that reproduction rate of predators is proportional to the number of prey consumed. This is like conversion of prey into new predators. 4. Beddington-DeAngelis type predator-prey models. Many predators compete for prey. This can result in time wasted in interfering each other’s effort of capturing and consuming prey. This consideration suggests that we may extend the principal of time budget to include predator interference time Ti . Let y be the predator density. It is plausible to think this interference time is proportional to the predator density y and the amount of prey captured Ha , but inversely proportional to the prey density. Hence Ti = byHa /x. Therefore Ha byHa T = Ts + Th + Ti = + Ha h + aρx x and hence aρxT Ha = . 1 + aρhx + abρy This gives the functional response of the form of aρx p(x, y) = . (4.1) 1 + aρhx + abρy The above functional response is called the Beddington-DeAngelis type functional response (Beddington 1975, DeAngelis et al. 1975). Predator-prey models employing such form of functional response are called Beddington-DeAngelis type predator-prey models. Some important Beddington-DeAngelis type predator-prey models are systematically studied by Hwang (2003, 2004). It is known that such models behalf more or less like the classical Holling type II predator-prey models. When aρhx >> 1 and abρy >> 1, then the above expression of p(x, y) can be approximated by aρx p(x, y) = . (4.2) aρhx + abρy This type of functional response is referred as (pure) ratio-dependent functional response. Predator-prey models employing such form of functional response are called ratio-dependent type predator-prey models.This maybe appropriate in situations where the predator-prey interaction takes place in an ever decreasing environment (patch size) while the predators are capable of searching a very large area in a unit of time such as eagles, tigers, wolves and whales. Ratio-dependent
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models are well-known for its ability to exhibit the often observed deterministic mutual extinctions of both prey and predator that often characterize the extinction of many predator and prey species (Kuang and Beretta 1998, Kuang 1999, Jost et al. 1999).
5. Stoichiometric population models. All organisms are composed of chemical elements such as carbon, nitrogen, and phosphorus. Although the relative abundance of these chemical constituents is known to vary considerably among species and across trophic levels, most ecological studies have until recently ignored the sources and consequences of this chemical heterogeneity. From theoretical perspectives, Lotka (1925) and other early researchers highlighted potential complications raised by having multiple currencies in ecological dynamics, but most subsequent work has focused instead on the dynamic implications of single currency (e.g., energy or carbon) models. However, rapidly accumulating evidence suggests that the dynamic implications of chemical heterogeneity among species deserve much more study than the subject has yet received. This body of research, which is to date chiefly empirical in nature, places major emphasis on the consequences of chemical heterogeneity among species for consumer-resource dynamics and nutrient recycling in ecosystems. Such multiple currency considerations enable simultaneous assessment of both food quantity and food quality. We refer to this approach as “ecological stoichiometry” (Sterner and Elser 2002). To complement and take advantage of the fast-growing empirical study of ecological stoichiometry, a variety of stoichiometry-based population models have been proposed and studied in recent years. These models vary from simple phenomenological two-dimensional resource-consumer models to more mechanistically formulated systems consisting of dozens of ordinary differential equations. In all these models, plant-herbivore interactions may shift from a (+, −) type to an unusual (−, −) class. This leads to dynamics with multiple equilibria, where bistability and deterministic extinction of the herbivore are possible. The most noteworthy dynamics is the birth of bistability as a result of large values of the carrying capacity K, which divides the plant-herbivore phase plane into two regions: one region with low-density but good-quality plants that sustain high-densities of herbivores, the other region with high density but low-quality plants that can sustain only low densities of herbivores(Loladze et al. 2000). In general, expressing plant-herbivore interactions in stoichiometrically realistic terms reveals qualitatively new dynamical behavior. The purpose of this section is to mention a simple, mathematically tractable model that provides a more mechanistic interpretation of the dynamics of plantherbivore interactions in a phosphorus (P )-limited environment. The key to our approach is the employment of variability in the P content of the plants, using the Droop equation for the plant’s growth. Our model takes the simple form of a system of two autonomous ordinary differential equations. It can be shown that the model of Loladze, Kuang, and Elser (2000), which we shall henceforth call the LKE model, is simply a special case of our model. To aid our model formulation and its comparison with the LKE model, it is convenient to recall here the main LKE model assumptions. They are A1. The total mass of phosphorus Pt in the entire system is fixed; i.e., the system is closed for phosphorus.
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A2. Phosphorus to carbon ratio (P :C) in the plant varies, but it never falls below a minimum q (mg P/mg C); the herbivore maintains a constant P:C ratio, denoted by θ (mg P/mg C). A3. All phosphorus in the system is divided into two pools: phosphorus in the herbivore and phosphorus in the plant. Assumption (A3) essentially assumes that free phosphorus is immediately taken by the plant. The LKE model takes the relatively simple form µ ¶ dx x = bx 1 − − f (x)y, dt θy)/q) µ min(K, (Pt − ¶ (LKE model) dy (Pt − θy)/x = e min 1, f (x)y − dy. dt θ where x is the density of plant (in milligrams of carbon per liter, mg C/l); y is the density of herbivore (mg C/l); b is the intrinsic growth rate of plant (day−1 ); d is the specific loss rate of herbivore that includes metabolic losses (respiration) and death (day−1 ); e is a constant production efficiency (yield constant); K is the plant’s constant carrying capacity that depends on some external factors such as light intensity; f (x) is the herbivore’s ingestion rate, which may be a Holling type II functional response. Note that µ ¶ µ ¶ x x x bx 1 − = bx min 1 − , 1 − min(K, (P − θy)/q) K (P − θy)/q and
µ bx 1 −
x (P − θy) /q
¶
µ
q = bx 1 − (P − θy) /x
¶ .
(5.1)
The left-hand side of (5.1) is a logistic equation, where (P − θy)/q is the carrying capacity of the plant determined by phosphorus availability. The right-hand side shows that it can be viewed as Droop’s equation (Droop 1973), where q is the minimal phosphorus content of the plant and (P − θy)/x is its actual phosphorus content. The LKE model can be interpreted as a special limiting case of a mechanistically formulated stoichiometric model (Kuang et al. 2004). The assumption (A3) of the LKE model stipulates that all phosphorus is in the herbivores and plants; that is, the concentration of freely available phosphorus (Pf ) is zero. This is tantamount to saying that the phosphorus uptake rate of the plants is extremely efficient. That is, α = ∞ or, equivalently α−1 = 0. If, in addition, we assume that the plant death rate D is negligibly small compared to its maximal growth rate, we may approximate the value of (µm − D)/µm as 1. Indeed, if we assume α → ∞ and (µm − D)/µm ≈ 1, the mechanistically formulated stoichiometric model (Kuang et al. 2004). mechanistic model simplifies to the LKE form. Acknowledgments. The research of Yang Kuang is supported in part by DMS0436341 and DMS/NIGMS-0342388.
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[29] Rosenzweig, M.L. (1971) Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science, 171, 385-387. [30] Sharov, A. (1996). Quantitative Population Biology, on-line lectures, Virginia Tech, http://www.ento.vt.edu/ sharov/PopEcol/popecol.html. [31] Sterner, R.W. and Elser J.J. (2002) Ecological Stoichiometry. Princeton University, Princeton, NJ. [32] Urabe, J. and R.W. Sterner (1996). Regulation of herbivore growth by the balance of light and nutrients. Proc. Natl. Acad. Sci. USA 93, 8465–8469. [33] Volterra, V. (1926) Fluctuations in the abandance of a species considered mathematically. Nature, 118, 558-60. E-mail address:
[email protected] 
