Lotka–Volterra方程式

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此词条Jie翻译。 由CecileLi初步审校,由潮升阶精校

卖给哈德逊湾公司的雪鞋野兔毛皮(后方黄色区域)和加拿大山猫毛皮(前方黑色线条)。注:加拿大山猫会以雪兔为食。

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

Lotka–Volterra方程式(又称为捕食者-猎物方程,以下简称为捕猎方程)是一对一阶非线性微分方程组,我们经常用它来描述两个物种间因捕食和被捕食关系而形成的动力学系统,这样的系统可称为捕猎系统。该方程组反映了此二者物种的种群数量会随时间变化并遵循如下规律:


[math]\displaystyle{ \begin{align} \frac{dx}{dt} &= \alpha x - \beta x y, \\ \frac{dy}{dt} &= \delta x y - \gamma y, \end{align} }[/math]


Where,

x is the number of prey (for example, rabbits);
y is the number of some predator (for example, foxes);
[math]\displaystyle{ \tfrac{dy}{dt} }[/math] and [math]\displaystyle{ \tfrac{dx}{dt} }[/math] represent the instantaneous growth rates of the two populations;
t represents time;
α, β, γ, δ are positive real parameters describing the interaction of the two species.

其中:

x是猎物(例如兔子)的数量;
y是捕食者(例如狐狸)的数量;
[math]\displaystyle{ \tfrac{dy}{dt} }[/math][math]\displaystyle{ \tfrac{dx}{dt} }[/math]代表个体的瞬时增长率;
t代表时间;
α, β, γ, δ是描述两个物种相互作用的正实参数。


The Lotka–Volterra system of equations is an example of a Kolmogorov model,[1][2][3] which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism.

Lotka–Volterra方程组是Kolmogorov模型的一个示例,而Kolmogorov模型[1][2][3] 具有更一般的模型框架,可以用来刻画捕食者与猎物之间因猎食,竞争,疾病和共生等关系而形成的生态动力系统。

历史

The Lotka–Volterra predator–prey model was initially proposed by Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910.[4][5] This was effectively the logistic equation,[6] originally derived by Pierre François Verhulst.[7] In 1920 Lotka extended the model, via Andrey Kolmogorov, to "organic systems" using a plant species and a herbivorous animal species as an example[8] and in 1925 he used the equations to analyse predator–prey interactions in his book on biomathematics.[9] The same set of equations was published in 1926 by Vito Volterra, a mathematician and physicist, who had become interested in mathematical biology.[5][10][11] Volterra's enquiry was inspired through his interactions with the marine biologist Umberto D'Ancona, who was courting his daughter at the time and later was to become his son-in-law. D'Ancona studied the fish catches in the Adriatic Sea and had noticed that the percentage of predatory fish caught had increased during the years of World War I (1914–18). This puzzled him, as the fishing effort had been very much reduced during the war years. Volterra developed his model independently from Lotka and used it to explain d'Ancona's observation.[12]

Lotka–Volterra捕猎模型最初是由阿尔弗雷德·J·洛特卡(Alfred J. Lotka)于1910年在自催化化学反应理论中提出的[4][5] 。这个模型实际上是一类逻辑方程[6] ,源自于皮埃尔·弗朗索瓦·韦吕勒(Pierre François Verhulst)[7] 。1920年,洛特卡以植食和草食性动物为例[8],在安德雷·柯尔莫哥洛夫(Andrey Kolmogorov)的帮助下将该模型扩展到了“有机系统”,并于1925年,他在自己编写的生物数学书中使用了这些方程式分析了捕食者与猎物之间的相互关系[9] 。1926年,数学和物理学家维托·沃尔泰拉(Vito Volterra)发表了同样的方程组。沃尔泰拉对数理生物学非常感兴趣[5][10][11] ,他对该领域的研究受到了与海洋生物学家翁贝托·德安科纳(Umberto D'Ancona)交流的启发,当时德安科纳正向他的女儿求婚,不久后便成了他的女婿。德安科纳研究了亚得里亚海的渔获物,并注意到在第一次世界大战期间(1914-1918年),捕捞的肉食性鱼类的百分比有所增加。因为这种现象恰好发生在捕鱼量已大大减少的战争年代,这使他感到困惑不已。后来,沃尔泰拉独立于洛特卡发展了自己的模型,并用它来解释德安科纳的观察结果[12]


The model was later extended to include density-dependent prey growth and a functional response of the form developed by C. S. Holling; a model that has become known as the Rosenzweig–MacArthur model.[13] Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey, such as the lynx and snowshoe hare data of the Hudson's Bay Company[14] and the moose and wolf populations in Isle Royale National Park.[15]

之后该模型得到进一步扩展,加入了包括依赖于种群密度的猎物增长机制和由霍林(C. S. Holling)发展出的功能响应机制,被称为 Rosenzweig–MacArthur模型[13] 。Lotka–Volterra和Rosenzweig–MacArthur模型一直被用于解释捕猎双方自然种群的动态变化,例如哈德逊湾[14]的山猫和雪兔的种群数据,以及罗亚尔岛国家公园[15]的麋鹿和狼的种群数据。


In the late 1980s, an alternative to the Lotka–Volterra predator–prey model (and its common-prey-dependent generalizations) emerged, the ratio dependent or Arditi–Ginzburg model.[16] The validity of prey- or ratio-dependent models has been much debated.[17]

在1980年代末,出现了Lotka–Volterra捕猎模型(泛指常规猎物依赖模型)的替代模型,即比率依赖模型或 Arditi–Ginzburg模型[16]。但时至今日,猎物依赖和比率依赖模型的有效性一直存在争议[17]


The Lotka–Volterra equations have a long history of use in economic theory; their initial application is commonly credited to Richard Goodwin in 1965[18] or 1967.[19][20]

Lotka-Volterra方程在理论经济学中有很长的应用历史,最早由理查德·古德温(Richard Goodwin)应用于1965[18] 与1967年[19][20]

方程的物理意义

The Lotka–Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:[21]

Lotka–Volterra模型对捕猎双方的环境和种群数量演化做出了许多假设,这些假设显得过于理想化,在自然界中显得并不实际[21]


  • The prey population finds ample food at all times.
  • The food supply of the predator population depends entirely on the size of the prey population.
  • The rate of change of population is proportional to its size.
  • During the process, the environment does not change in favour of one species, and genetic adaptation is inconsequential.
  • Predators have limitless appetite.
  • 猎物随时都有充足的食物。
  • 捕食者种群的食物供应充足与否完全取决于猎物种群的大小。
  • 各种群数量变化率与其规模成正比。
  • 在此过程中,环境不会因一种物种而改变,并且忽略遗传适应性。
  • 捕食者有无限的食欲。


In this case the solution of the differential equations is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[22]

在这种情况下,微分方程的解是确定并连续的。反过来,这也意味着捕食者和猎物的世代是持续重叠的[22]


猎物

When multiplied out, the prey equation becomes

当猎物数量成倍增趋势时,猎物方程可表示为:


[math]\displaystyle{ \frac{dx}{dt} = \alpha x - \beta x y. }[/math]


The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet, this is represented above by βxy. If either x or y is zero, then there can be no predation.

假定猎物具有无限的食物供应,且除非受到捕食,否则可以成倍繁殖,那么其指数增长可表示为上式中的αx。假设捕食者的捕食率,与捕食者和猎物的相遇率成正比,那么即可表示为上式中的βxy。注意如果xy为零,则表示没有发生捕猎。


With these two terms the equation above can be interpreted as follows: the rate of change of the prey's population is given by its own growth rate minus the rate at which it is preyed upon.

基于这两个假设,上述方程式可以理解为:猎物种群的变化率等于其自身数量的增长率减去被捕食数量的增长率。


捕食者

The predator equation becomes

捕食者方程可表示为:


[math]\displaystyle{ \frac{dy}{dt} = \delta xy - \gamma y. }[/math]


In this equation, δxy represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used, as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). The term γy represents the loss rate of the predators due to either natural death or emigration, it leads to an exponential decay in the absence of prey.

在此等式中,δxy代表捕食者种群的增长。(请注意此处与捕食率表达式虽然相似;但是使用了一个不同的常数,因为捕食者的增长速率不一定等于其捕杀猎物的速率)。另外γy表示由于自然死亡或迁徙造成的捕食者数量减少率,它在没有猎物的情况下是呈指数型衰减的。


Hence the equation expresses that the rate of change of the predator's population depends upon the rate at which it consumes prey, minus its intrinsic death rate.

因此,上述方程式可以理解为,捕食者种群的变化率取决于其捕杀猎物的速率减去其内在死亡(包括迁徙)率。

方程求解

The equations have periodic solutions and do not have a simple expression in terms of the usual trigonometric functions, although they are quite tractable.[23][24]

对于通常的三角方程,虽然很容易处理并得到其周期解,但是其解的表达式并不简洁[23][24]


If none of the non-negative parameters α, β, γ, δ vanishes, three can be absorbed into the normalization of variables to leave only one parameter: since the first equation is homogeneous in x, and the second one in y, the parameters β/α and δ/γ are absorbable in the normalizations of y and x respectively, and γ into the normalization of t, so that only α/γ remains arbitrary. It is the only parameter affecting the nature of the solutions.

如果该方程组中所有非负参数α, β, γ, δ均存在,那么我们可以将其中三个变量进行归一化,仅留下一个参数:由于第一个方程对于x而言是齐次的,且第二个方程对于y而言也是齐次的,因此我们可以分别对yx的参数归一化处理为β/αδ/γ,再令γt作归一化,最后只有α/γ保持任意,它成为了影响解的性质的唯一参数。


A linearization of the equations yields a solution similar to simple harmonic motion[25] with the population of predators trailing that of prey by 90° in the cycle.

方程的线性化后求得的解类似于简谐运动曲线[25] ,捕食者的数量比猎物的数量落后大概四分之一个周期。


Linearized version

模板:Further

模板:Anchor


简单示例

前文提到关于狒狒和猎豹问题的种群动态。
当捕食者种群在不同初始条件下,其捕猎问题的相空间图。

Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 10 baboons and 10 cheetahs, one can plot the progression of the two species over time; given the parameters that the growth and death rates of baboon are 1.1 and 0.4 while that of cheetahs are 0.1 and 0.4 respectively. The choice of time interval is arbitrary.

假设有两种动物,狒狒(猎物)和猎豹(捕食者)。如果初始条件是10只狒狒和10只猎豹,狒狒的增长率和死亡率分别为1.1和0.4,而猎豹的增长率和死亡率分别为0.1和0.4,且在时间轴上从初始点开始任选一段足够长的区间,那么我们可以绘制出这两个物种随时间推移的数量变化曲线。


One may also plot solutions parametrically as orbits in phase space, without representing time, but with one axis representing the number of prey and the other axis representing the number of predators for all times.

或者也可以在相空间轨道中将其解进行参数化处理,此时就可以略去时间轴。仅用其中一个轴代表全时间段猎物的数量,而另一轴代表全时间段捕食者的数量。


This corresponds to eliminating time from the two differential equations above to produce a single differential equation

对应于上面的两个微分方程,此方法可以得出约掉时间参数的一个全新的微分方程


[math]\displaystyle{ \frac{dy}{dx} = - \frac{y}{x} \frac{\delta x - \gamma}{\beta y -\alpha} }[/math]


relating the variables x and y. The solutions of this equation are closed curves. It is amenable to separation of variables: integrating

该方程仅包含变量xy。方程的解是一条闭合曲线,可以通过分离变量法求解:对以下式子进行积分


[math]\displaystyle{ \frac{\beta y - \alpha}{y} \,dy + \frac{\delta x - \gamma}{x} \,dx = 0 }[/math]


yields the implicit relationship

得到隐性关系


[math]\displaystyle{ V = \delta x - \gamma \ln(x) + \beta y - \alpha \ln(y), }[/math]


where V is a constant quantity depending on the initial conditions and conserved on each curve.

其中V是取决于初始条件的常量,并且在每条曲线上均守恒。


An aside: These graphs illustrate a serious potential problem with this as a biological model: For this specific choice of parameters, in each cycle, the baboon population is reduced to extremely low numbers, yet recovers (while the cheetah population remains sizeable at the lowest baboon density). In real-life situations, however, chance fluctuations of the discrete numbers of individuals, as well as the family structure and life-cycle of baboons, might cause the baboons to actually go extinct, and, by consequence, the cheetahs as well. This modelling problem has been called the "atto-fox problem", an atto-fox being a notional 10−18 of a fox.[26][27]

另外值得注意的是,这些图说明了该方程组作为生物学模型的严重潜在问题:因为某种特定的参数选择,在每个周期中,狒狒的数量都被减少到极低的数量,但却有能力恢复(并且在极低的狒狒种群密度之下,猎豹的数量仍然很大)。这在现实中显然是不可能的,离群索居的狒狒的数量的波动,以及狒狒的家庭结构和生命周期都有可能导致狒狒种族灭绝,结果也就造成了猎豹的灭绝。按照此类方法建模出现的问题被称为"atto-fox问题","atto"这里指的是十的负十八次方[26][27]


相空间图的进一步示例

捕猎相空间图

A less extreme example covers: α = 2/3, β = 4/3, γ = 1 = δ. Assume x, y quantify thousands each. Circles represent prey and predator initial conditions from x = y = 0.9 to 1.8, in steps of 0.1. The fixed point is at (1, 1/2).

一个较为合理的例子是: α = 2/3, β = 4/3, γ = 1 = δ. 假设x, y处于“千”级别还不到“万“。圆圈代表从x = y = 0.9 到 1.8时猎物和捕食者的初始条件,步长为0.1,不动点为(1,1/2)。

系统动力学

In the model system, the predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and decline. As the predator population is low, the prey population will increase again. These dynamics continue in a cycle of growth and decline.

在该模型系统中,捕食者在有大量猎物的情况下肆意增长,但最终它们会因为食物供应不足而下降。随即捕食者数量变低,猎物数量将再次增加。这样形成的动力学系统以增长和下降为周期持续下去。


种群平衡

Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the derivatives are equal to 0:

当捕猎双方种群数量都没有变化时,即当式中两个导数都等于0时,模型就会出现种群平衡:


[math]\displaystyle{ x(\alpha - \beta y) = 0, }[/math]
[math]\displaystyle{ -y(\gamma - \delta x) = 0. }[/math]


The above system of equations yields two solutions:

对上面的方程组进行求解,得到:


[math]\displaystyle{ \{y = 0,\ \ x = 0\} }[/math]


and


[math]\displaystyle{ \left\{y = \frac{\alpha}{\beta},\ \ x = \frac{\gamma}{\delta} \right\}. }[/math]


Hence, there are two equilibria.

因此我们得到了两个平衡点(对应于上式的两个解)。


The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters α, β, γ, and δ.

第一种解实际上代表了两种物种的灭绝。当它们的解都为0时,它们将继续无限期地保持灭绝状态。第二种解则表示两个种群都维持其当前固定数量(非零点),在简化模型中,它们无限期地保持状态不变。达到平衡时的种群数量水平取决于参数α, β, γ, 和 δ的给定值。


不动点的稳定性

The stability of the fixed point at the origin can be determined by performing a linearization using partial derivatives.

不动点在原点处的稳定性可以通过求偏导数将其线性化来确定。


The Jacobian matrix of the predator–prey model is

可以得出捕猎模型的雅可比矩阵为:


[math]\displaystyle{ J(x, y) = \begin{bmatrix} \alpha - \beta y & -\beta x \\ \delta y & \delta x - \gamma \end{bmatrix}. }[/math]


and is known as the community matrix.

该矩阵也可以称为群落矩阵。


第一不动点(灭绝)

When evaluated at the steady state of (0, 0), the Jacobian matrix J becomes

当在(0, 0)的稳态下求值时,雅可比矩阵J


[math]\displaystyle{ J(0, 0) = \begin{bmatrix} \alpha & 0 \\ 0 & -\gamma \end{bmatrix}. }[/math]


The eigenvalues of this matrix are

该矩阵的特征值是:


[math]\displaystyle{ \lambda_1 = \alpha, \quad \lambda_2 = -\gamma. }[/math]


In the model α and γ are always greater than zero, and as such the sign of the eigenvalues above will always differ. Hence the fixed point at the origin is a saddle point.

在模型中,由于αγ始终大于零,因此上述特征值的符号将始终不同。继而得到原点的不动点是鞍点。


The stability of this fixed point is of significance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, it follows that the extinction of both species is difficult in the model. (In fact, this could only occur if the prey were artificially completely eradicated, causing the predators to die of starvation. If the predators were eradicated, the prey population would grow without bound in this simple model.) The populations of prey and predator can get infinitesimally close to zero and still recover.

第一不动点的稳定性非常重要。只有当它是稳定的,非零物种群体才有可能趋向于灭绝。并且在此条件下,即使种群处于初期状态,根据该动力学系统仍然可能得出两种物种都灭绝的结果。但是,由于第一不动点位于原点处是一个鞍点,因此很不稳定,进而说明模型中的两个物种都灭绝并没有那么容易。(实际上,只有在人为地彻底消灭猎物,进而导致捕食者饿死之后,才会发生这种情况。相反地,如果消灭了捕食者,那么在这个简单的模型中,猎物将不受限制地增长。)捕食者和猎物的种群数量可以无限地接近零,且仍然可以恢复。


第二不动点(震荡)

Evaluating J at the second fixed point leads to

在第二个不动点求雅可比矩阵J可得:


[math]\displaystyle{ J\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) = \begin{bmatrix} 0 & -\frac{\beta \gamma}{\delta} \\ \frac{\alpha \delta}{\beta} & 0 \end{bmatrix}. }[/math]


The eigenvalues of this matrix are

该矩阵的特征值是


[math]\displaystyle{ \lambda_1 = i \sqrt{\alpha \gamma}, \quad \lambda_2 = -i \sqrt{\alpha \gamma}. }[/math]


As the eigenvalues are both purely imaginary and conjugate to each others, this fixed point is elliptic, so the solutions are periodic, oscillating on a small ellipse around the fixed point, with a frequency [math]\displaystyle{ \omega = \sqrt{\lambda_1 \lambda_2} = \sqrt{\alpha \gamma} }[/math] and period [math]\displaystyle{ T = 2{\pi}/(\sqrt{\lambda_1 \lambda_2}) }[/math].

由于特征值既是纯虚数又是共轭的,因此该不动点是椭圆的并且其解具有周期性,即在不动点周围的椭圆环上以一定频率振荡:[math]\displaystyle{ \omega = \sqrt{\lambda_1 \lambda_2} = \sqrt{\alpha \gamma} }[/math] ,其周期为:[math]\displaystyle{ T = 2{\pi}/(\sqrt{\lambda_1 \lambda_2}) }[/math]


As illustrated in the circulating oscillations in the figure above, the level curves are closed orbits surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate without damping around the fixed point with frequency [math]\displaystyle{ \omega = \sqrt{\alpha \gamma} }[/math].

如上图的循环振荡所示,其等高线围绕不动点形成闭合轨道:因此捕食者和猎物的种群数量在不动点处以频率[math]\displaystyle{ \omega = \sqrt{\alpha \gamma} }[/math]循环并且无阻尼震荡。


The value of the constant of motion V, or, equivalently, K = exp(V), [math]\displaystyle{ K = y^\alpha e^{-\beta y} x^\gamma e^{-\delta x} }[/math], can be found for the closed orbits near the fixed point.

对于不动点附近的闭合轨道,不难找到其运动常数V的值,同样包括,K = exp(V), [math]\displaystyle{ K = y^\alpha e^{-\beta y} x^\gamma e^{-\delta x} }[/math]


Increasing K moves a closed orbit closer to the fixed point. The largest value of the constant K is obtained by solving the optimization problem

K的增加会将闭合轨道移近不动点。另外通过解决优化问题还可以获得常数K的最大值。


[math]\displaystyle{ y^\alpha e^{-\beta y} x^\gamma e^{-\delta x} = \frac{y^\alpha x^\gamma}{e^{\delta x+\beta y}} \longrightarrow \max\limits_{x,y\gt 0}. }[/math]


The maximal value of K is thus attained at the stationary (fixed) point [math]\displaystyle{ \left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) }[/math] and amounts to

因此,在固定点(即不动点)[math]\displaystyle{ \left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) }[/math]处可以获得K的最大值,该值等于[math]\displaystyle{ K^* = \left(\frac{\alpha}{\beta e}\right)^\alpha \left(\frac{\gamma}{\delta e}\right)^\gamma, }[/math]


where e is Euler's number.

其中e是欧拉数

另请参见


  • 竞争性Lotka–Volterra方程Competitive Lotka–Volterra equations
  • 广义Lotka–Volterra方程Generalized Lotka–Volterra equation
  • 互惠主义与Lotka–Volterra方程Mutualism and the Lotka–Volterra equation
  • 群落矩阵Community matrix
  • 种群动态Population dynamics
  • 渔业种群动态Population dynamics of fisheries
  • Nicholson–Bailey模型Nicholson–Bailey model
  • 反应扩散系统Reaction–diffusion system
  • 富集悖论Paradox of enrichment
  • 兰切斯特定律,有关于军事战术的微分方程组Lanchester's laws, a similar system of differential equations for military forces

备注

  1. 1.0 1.1 Freedman, H. I. (1980). Deterministic Mathematical Models in Population Ecology. Marcel Dekker. 
  2. 2.0 2.1 Brauer, F.; Castillo-Chavez, C. (2000). Mathematical Models in Population Biology and Epidemiology. Springer-Verlag. 
  3. 3.0 3.1 Hoppensteadt, F. (2006). "Predator-prey model". Scholarpedia. 1 (10): 1563. Bibcode:2006SchpJ...1.1563H. doi:10.4249/scholarpedia.1563.
  4. 4.0 4.1 Lotka, A. J. (1910). "Contribution to the Theory of Periodic Reaction". J. Phys. Chem. 14 (3): 271–274. doi:10.1021/j150111a004.
  5. 5.0 5.1 5.2 5.3 Goel, N. S. (1971). On the Volterra and Other Non-Linear Models of Interacting Populations. Academic Press. 
  6. 6.0 6.1 Berryman, A. A. (1992). "The Origins and Evolution of Predator-Prey Theory" (PDF). Ecology. 73 (5): 1530–1535. doi:10.2307/1940005. JSTOR 1940005. Archived from the original (PDF) on 2010-05-31.
  7. 7.0 7.1 Verhulst, P. H. (1838). "Notice sur la loi que la population poursuit dans son accroissement". Corresp. Mathématique et Physique. 10: 113–121.
  8. 8.0 8.1 Lotka, A. J. (1920). "Analytical Note on Certain Rhythmic Relations in Organic Systems". Proc. Natl. Acad. Sci. U.S.A. 6 (7): 410–415. Bibcode:1920PNAS....6..410L. doi:10.1073/pnas.6.7.410. PMC 1084562. PMID 16576509.
  9. 9.0 9.1 Lotka, A. J. (1925). Elements of Physical Biology. Williams and Wilkins. 
  10. 10.0 10.1 Volterra, V. (1926). "Variazioni e fluttuazioni del numero d'individui in specie animali conviventi". Mem. Acad. Lincei Roma. 2: 31–113.
  11. 11.0 11.1 Volterra, V. (1931). "Variations and fluctuations of the number of individuals in animal species living together". In Chapman, R. N.. Animal Ecology. McGraw–Hill. 
  12. 12.0 12.1 Kingsland, S. (1995). Modeling Nature: Episodes in the History of Population Ecology. University of Chicago Press. ISBN 978-0-226-43728-6. 
  13. 13.0 13.1 Rosenzweig, M. L.; MacArthur, R.H. (1963). "Graphical representation and stability conditions of predator-prey interactions". American Naturalist. 97 (895): 209–223. doi:10.1086/282272. Unknown parameter |s2cid= ignored (help)
  14. 14.0 14.1 Gilpin, M. E. (1973). "Do hares eat lynx?". American Naturalist. 107 (957): 727–730. doi:10.1086/282870. Unknown parameter |s2cid= ignored (help)
  15. 15.0 15.1 Jost, C.; Devulder, G.; Vucetich, J.A.; Peterson, R.; Arditi, R. (2005). "The wolves of Isle Royale display scale-invariant satiation and density dependent predation on moose". J. Anim. Ecol. 74 (5): 809–816. doi:10.1111/j.1365-2656.2005.00977.x.
  16. 16.0 16.1 Arditi, R.; Ginzburg, L. R. (1989). "Coupling in predator-prey dynamics: ratio dependence" (PDF). Journal of Theoretical Biology. 139 (3): 311–326. doi:10.1016/s0022-5193(89)80211-5.
  17. 17.0 17.1 Abrams, P. A.; Ginzburg, L. R. (2000). "The nature of predation: prey dependent, ratio dependent or neither?". Trends in Ecology & Evolution. 15 (8): 337–341. doi:10.1016/s0169-5347(00)01908-x. PMID 10884706.
  18. 18.0 18.1 Gandolfo, G. (2008). "Giuseppe Palomba and the Lotka–Volterra equations". Rendiconti Lincei. 19 (4): 347–357. doi:10.1007/s12210-008-0023-7. Unknown parameter |s2cid= ignored (help)
  19. 19.0 19.1 Goodwin, R. M. (1967). "A Growth Cycle". In Feinstein, C. H.. Socialism, Capitalism and Economic Growth. Cambridge University Press. https://archive.org/details/socialismcapital0000fein. 
  20. 20.0 20.1 Desai, M.; Ormerod, P. (1998). "Richard Goodwin: A Short Appreciation" (PDF). The Economic Journal. 108 (450): 1431–1435. CiteSeerX 10.1.1.423.1705. doi:10.1111/1468-0297.00350. Archived from the original (PDF) on 2011-09-27. Retrieved 2010-03-22.
  21. 21.0 21.1 "PREDATOR-PREY DYNAMICS". www.tiem.utk.edu. Retrieved 2018-01-09.
  22. 22.0 22.1 Cooke, D.表达式错误:无法识别的词语“etal”。 (1981). The Mathematical Theory of the Dynamics of Biological Populations. II. Academic Press. 
  23. 23.0 23.1 Steiner, Antonio; Gander, Martin Jakob (1999). "Parametrische Lösungen der Räuber-Beute-Gleichungen im Vergleich". Il Volterriano. 7: 32–44.
  24. 24.0 24.1 Evans, C. M.; Findley, G. L. (1999). "A new transformation for the Lotka-Volterra problem". Journal of Mathematical Chemistry. 25: 105–110. doi:10.1023/A:1019172114300. Unknown parameter |s2cid= ignored (help)
  25. 25.0 25.1 Tong, H. (1983). Threshold Models in Non-linear Time Series Analysis. Springer–Verlag. 
  26. 26.0 26.1 Lobry, Claude; Sari, Tewfik (2015). "Migrations in the Rosenzweig-MacArthur model and the "atto-fox" problem" (PDF). Arima. 20: 95–125.
  27. 27.0 27.1 Mollison, D. (1991). "Dependence of epidemic and population velocities on basic parameters" (PDF). Math. Biosci. 107 (2): 255–287. doi:10.1016/0025-5564(91)90009-8. PMID 1806118.

参考文献

  • Leigh, E. R. (1968). "The ecological role of Volterra's equations". Some Mathematical Problems in Biology.  – a modern discussion using Hudson's Bay Company data on lynx and hares in Canada from 1847 to 1903.
  • Murray, J. D. (2003). Mathematical Biology I: An Introduction. New York: Springer. ISBN 978-0-387-95223-9. 

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