LPA Model Simulator

This website provides interactive simlations for the discrete-time LPA (larva-pupa-adult) population model. Use the links on menu to the left to navigate to the different simulation views. Below is a descriptions of the equations and simulation views for the basic LPA model.

LPA Model

The LPA model describes the population dynamics of flour beetles. The model given by the following three equations: \begin{align} L_{t+1} & = b A_t \exp\bigl(-c_{el} L_t - c_{ea} A_t\bigr), \\ P_{t+1} & = L_t \bigl(1-\mu_l\bigr), \\ A_{t+1} & = P_t \exp\bigl(-c_{pa} A_t\bigr) + A_t \bigl(1-\mu_a\bigr). \end{align} The first equation is for the number of feeding larvae (L variable), the second is for the number of large larvae, non-feeding larvae, pupae and callow adults (P variable), and the third is for the number of sexually mature adults (A variable). The unit of time is two weeks and is, approximately, the average amount of time spent in the feeding larval stage under experimental conditions. The time unit is also approximately the average duration of the P-stage. The quantity \(b > 0\) is the number of larval recruits per adult per unit of time in the absence of cannibalism. The fractions \(\mu_l\) and \(\mu_a\) are the larval and adult rates of mortality in one time unit. The exponential functions account for the cannibalism of eggs by both larvae and adults and the cannibalism of pupae by adults. The fractions \(\exp\bigl(-c_{el} L(t)\bigr)\) and \(\exp\bigl(-c_{ea} A(t)\bigr)\) are the probabilities that an egg is not eaten in the presence of \(L_t\) larvae and \(A_t\) adults in one time unit. The fraction \(\exp\bigl(-c_{pa} A_t\bigr)\) is the survival probability of a pupa in the presence of \(A_t\) adults in one time unit.

Simulation Views

The three simulation views are described below.

Time series. This view is a plot of the numbers of the larvae, pupae, and adults over time. You can hide/show the plots for each life stage by clicking on its legend entry.

State space. This view is a three-dimensional plot of the larvae, pupae, and adults, where the three axes are the values for the L, P, and A state variables. A time series is created and the L, P, and A values are plotted. You have the option of discarding initial transient values as the system approaches the attractor. You also have the options of connecting the time series points with lines and changing the size of the plot symbols.

Bifurcation plot. In this view you can see how the LPA model attractor changes as a bifurcation parameter is varied. You have options for choosing the bifurcation parameter, its plot range, and the system variable that appears on the y-axis (larvae, pupae, adults, or total population size).

Links to each of these simulation views can be found at the top of the left column of this page.

Other Model Features

Other features of the LPA Model Simulator include the following:

Stochasticity. Stochastic versions of the LPA model can be simulated in the time series and state space views. (Bifiurcation plots are inherently deterministic.) Four types of stochastic models can be explored: (1) logarithmic-scale environmental noise, (2) square-root-scale demographic noise, (3) a Poisson-binomial model, and (4) a negative binomial model.

Lattice effects. The population dynamics can be constrained to an interger-based state space lattice. Different algorithms can be chosen to accomplish this.

Habitat size. The size of the habitat (amount of flour medium) can be varied. Habtiat size can be kept constant or you can choose alternating habitat sizes.

Links to these simulation features are found in the right column of the time series and state space views.

The LPA Model Simulator can be used to explore examples covered in the book Complex Population Dynamics: Theory and Data (2025, CRC Press). There is a brief description of each example, a reference to its location in the book, and a link to load the parameter values and other settings into the simulator. Use the link at the bottom left side of this page to access the examples.

For more detailed information on the LPA Model Simulator, see the pdf manual. A link is provided in the middle of the left column of this page.

LPA Simulation Examples from
Complex Population Dynamics: Theory and Data

+ Chapter 5: The LPA Model

+ Chapter 6: Transitions between Attractors

+ §6.1 Bifurcations in the LPA model

+ §6.2 A bifurcation experiment

+ Chapter 7: The Hunt for Chaos

+ §7.1 A model predicted route-to-chaos

+ §7.4 Attractors in the route-to-chaos experiment

+ §7.5 The stochastic LPA model

Description:

Stochastic model orbits for the route-to-chaos experiment using the estimated CLS parameter values and variance-covariance matrices [3].

Parameter values and initial numbers:

\(b\)	\(c_{el}\)	\(c_{ea}\)	\(c_{pa}\)	\(\mu_l\)	\(\mu_a\)	\(L_0\)	\(P_0\)	\(A_0\)
10.45	0.01731	0.01310	0.004619	0.2000	0.007629	250	5	100

Stochasticity: demographic (square-root-scale) with variances and covariances

Group	\(\sigma_{11}\)	\(\sigma_{12}=\sigma_{21}\)	\(\sigma_{13}=\sigma_{31}\)	\(\sigma_{22}\)	\(\sigma_{23}=\sigma_{32}\)	\(\sigma_{33}\)
Control	1.621	−0.1336	−0.01339	0.7375	−0.0009612	0.01212
Treatments	2.332	0.007097	0	0.2374	0	0

Lattice effects: none

Habitat size: constant at V = 1 (20 g medium)

Examples:

Stochastic with parameters for controls, time series and state space
Stochastic with \(c_{pa}=0.00\) and \(\mu_a=0.96\), time series and state space
Stochastic with \(c_{pa}=0.05\) and \(\mu_a=0.96\), time series and state space
Stochastic with \(c_{pa}=0.10\) and \(\mu_a=0.96\), time series and state space
Stochastic with \(c_{pa}=0.25\) and \(\mu_a=0.96\), time series and state space
Stochastic with \(c_{pa}=0.35\) and \(\mu_a=0.96\), time series and state space
Stochastic with \(c_{pa}=0.50\) and \(\mu_a=0.96\), time series and state space
Stochastic with \(c_{pa}=1.00\) and \(\mu_a=0.96\), time series and state space

+ §7.6 The chaotic attractor

+ Chapter 8: Lattice Effects

+ §8.3 Beetles on a lattice

+ §8.4 Constructing lattice models

+ Chapter 9: How Is Chaos Manifest in Noisy Populations?

+ Chapter 10: Demographic Stochasticity

+ §10.5 Example using the LPA model

Description:

Simulations showing how the influence of demographic stochasticity wanes as habitat size increases [7]. ML parameter estimates are from the Hunt for Chaos experiment [3].

Parameter values and initial numbers:

\(b\)	\(c_{el}\)	\(c_{ea}\)	\(c_{pa}\)	\(\mu_l\)	\(\mu_a\)	\(L_0\)	\(P_0\)	\(A_0\)
10.67	0.01647	0.01313	0.35	0.1955	0.96	250	5	100

Stochasticity: deterministic, Poisson-binomial, and negative binomial models

Lattice effects: none, Poisson-binomial, and negative binomial

Habitat size: constant at V = 1, 3, 10, 100 (20, 60, 200, 2000 g medium)

Examples:

The deterministic chaotic attractor, time series and state space
Poisson-binomial with V = 1, time series and state space
Poisson-binomial with V = 3, time series and state space
Poisson-binomial with V = 10, time series and state space
Poisson-binomial with V = 100, time series and state space
Negative binomial with \(\sigma^2 = 5\) and V = 1, time series and state space
Negative binomial with \(\sigma^2 = 5\) and V = 10, time series and state space
Negative binomial with \(\sigma^2 = 5\) and V = 100, time series and state space
Negative binomial with \(\sigma^2 = 10\) and V = 1, time series and state space
Negative binomial with \(\sigma^2 = 10\) and V = 10, time series and state space
Negative binomial with \(\sigma^2 = 10\) and V = 100, time series and state space

+ Chapter 13: Cycles and Phase Switching

+ References

[1]	Dennis B, Desharnais RA, Cushing JM, Costantino RF (1995) Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments. Ecological Monographs 65:261–282.
[2]	Dennis B, Desharnais RA, Cushing JM, Costantino RF (1997) Transitions in population dynamics: equilibria to periodic cycles to aperiodic cycles. Journal of Animal Ecology 66:704–729.
[3]	Dennis B, Desharnais RA, Cushing JM, Henson SM, Costantino RF (2001) Estimating chaos and complex dynamics in an insect population. Ecological Monographs 71:277–303.
[4]	Henson SM, Costantino RF, Cushing JM, Desharnais RA, Dennis B, King AA (2001) Lattice effects observed in chaotic dynamics of experimental populations. Science 294:602–605.
[5]	Henson SM, King AA, Costantino RF, Cushing JM, Dennis B, Desharnais RA (2003) Explaining and predicting patterns in stochastic population systems. Proceedings of the Royal Society of London. Series B: Biological Sciences 270:1549–1553.
[6]	King AA, Costantino RF, Cushing JM, Henson SM, Desharnais RA, Dennis B (2004) Anatomy of a chaotic attractor: subtle model-predicted patterns revealed in population data. Proceedings of the National Academy of Sciences 101:408–413.
[7]	Desharnais RA, Costantino RF, Cushing JM, Henson SM, Dennis B, King AA (2006) Experimental support of the scaling rule for demographic stochasticity. Ecology Letters 9:537–547.
[8]	Henson SM, Cushing JM, Costantino RF, Dennis B, Desharnais RA (1998) Phase switching in population cycles. Proceedings of the Royal Society of London B 265:2229–2234.

\(\sigma_{11}\)	\(\sigma_{12}=\sigma_{21}\)	\(\sigma_{13}=\sigma_{31}\)	\(\sigma_{22}\)	\(\sigma_{23}=\sigma_{32}\)	\(\sigma_{33}\)
0.2771	0.02792	0.009796	0.4284	−0.008150	0.01779