循环的Lotka-Volterra竞争系统具有离散的时间延迟
Akio Matsumoto1, Ferenc Szidarovszky2
1Chuo University, Tokyo, Japan.
Nonlinear dynamics, psychology, and life sciences
|March 20, 2024
概括
这项研究分析了三种有延迟的Lotka-Volterra模型,发现稳定性损失导致限制周期. 延迟会影响物种活动,在由此产生的周期中,有两种物种活跃,一种不活跃.
科学领域:
- 生态生态学 生态生态学
- 数学生物学 数学生物学
- 动态系统 动态系统
背景情况:
- 洛特卡-沃尔特拉模型是研究物种相互作用的基本工具.
- 了解时间延迟对生态模型的影响对于现实的模拟至关重要.
- 以前的研究往往集中在更简单的模型或单一的延迟.
研究的目的:
- 为了研究一个具有两个离散时间延迟的三种Lotka-Volterra竞争模型的复杂动态.
- 为了确定正静止点的稳定性由于这些延迟而丧失的条件.
- 分析由此产生的分叉和物种活动模式.
主要方法:
- 对与延迟模型相关的立方指数多项式特征方程的分析.
- 构建稳定性切换曲线以确定关键延迟值.
- 数字模拟以确认Hopf分叉和限制循环行为.
- 检查沿着极限周期的物种活动,因为延迟有所不同.
主要成果:
- 确定了稳定性切换曲线,表明在特定延迟值时可能会失去稳定性.
- 证实了超临界的霍夫分叉,导致在延迟跨越稳定曲线时限制周期振荡.
- 随着延迟的增加,模型显示了一个模式,其中两个物种保持活跃,而一个物种在极限周期内变得不活跃.
结论:
- 离散的时间延迟可以破坏三种竞争性Lotka-Volterra模型中的正静止点的稳定性.
- 霍夫分叉是这些延迟系统中产生振荡动态的关键机制.
- 延迟的存在可能导致简化生态动态,一些物种在长期内变得不活跃.
相关概念视频
01:31Classification of Systems-II
145
Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
145
01:23Linear time-invariant Systems
257
A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
257
01:26Linear Approximation in Frequency Domain
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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
89
01:18Second Order systems II
109
In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
109
01:21Linear Approximation in Time Domain
81
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
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01:26Classification of Systems-I
184
Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
184