Alternative Stable States, Coral Reefs, and Smooth Dynamics with a Kick

Stephen Ippolito; Vincent Naudot; Erik G. Noonburg

doi:10.1007/s11538-016-0148-2

Bulletin of Mathematical Biology

March 2016, Volume 78, Issue 3, pp 413–435 | Cite as

Alternative Stable States, Coral Reefs, and Smooth Dynamics with a Kick

Authors
Authors and affiliations

Stephen IppolitoEmail author
Vincent Naudot
Erik G. Noonburg

Original Article

First Online: 07 March 2016

Received: 12 May 2015

Accepted: 15 February 2016

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Abstract

We consider a computer simulation, which was found to be faithful to time series data for Caribbean coral reefs, and an analytical model to help understand the dynamics of the simulation. The analytical model is a system of ordinary differential equations (ODE), and the authors claim this model demonstrates the existence of alternative stable states. The existence of an alternative stable state should consider a sudden shift in coral and macroalgae populations, while the grazing rate remains constant. The results of such shifts, however, are often confounded by changes in grazing rate. Although the ODE suggest alternative stable states, the ODE need modification to explicitly account for shifts or discrete events such as hurricanes. The goal of this paper will be to study the simulation dynamics through a simplified analytical representation. We proceed by modifying the original analytical model through incorporating discrete changes into the ODE. We then analyze the resulting dynamics and their bifurcations with respect to changes in grazing rate and hurricane frequency. In particular, a “kick” enabling the ODE to consider impulse events is added. Beyond adding a “kick” we employ the grazing function that is suggested by the simulation. The extended model was fit to the simulation data to support its use and predicts the existence cycles depending nonlinearly on grazing rates and hurricane frequency. These cycles may bring new insights into consideration for reef health, restoration and dynamics.

Keywords

Coral reefs Alternative stable states Dirac delta distribution Dynamical systems Competitive dynamics

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Appendix

In this paper, our goal was to represent a computer simulation, analytically, which has been shown to be successful at modeling coral reef dynamics. In doing so, our analysis of the simulation has suggested using a grazing term in the form of gM which can be thought of as a Type I functional response with a constant parrot fish population as the predator. Typically, a Type II functional response is thought to be more biologically accurate, so we have included model fitting and bifurcation results for (7) with gM replaced by $gM/(1+bM)$. This can be thought of as a Type II functional repose with a constant parrot fish population as the predator. Here we add the parameter b which is proportional to the handling time or the times it takes a parrot fish to consume a patch of macroalgae. The model is given in (22) below.

$$\begin{aligned} \left\{ \begin{array}{rcl} \displaystyle \frac{\hbox {d}M}{\hbox {d}t} &{} = &{} aMC - \frac{gM}{1+bM} +\gamma MT +\sum \limits _{n =0}^{\infty } h_1 (1-M) \delta (t-nk)\\ \displaystyle \frac{\hbox {d}C}{\hbox {d}t} &{} = &{} rTC - \zeta C - aMC + \sum \limits _{n =0}^{\infty } h_2 C \delta (t-nk) \end{array} \right. \end{aligned}$$

(22)

Type II Response Fits

Here we consider fitting (22) to the simulation data, similar to what was done for (7) in Fig. 4 and Table 3. If we first compare Figs. 4 and 8, we note that except for Trials 8 and 9 there does not appear to be much distinction between the model fits for the simulation data. Comparing Tables 3 and 4, we see a noticeable improvement for the $R^2$ value for Trials 1, 7 and 9. Also while there are stability issues fitting data for Trial 8 with (7), this problem did not persist with (22); however, the $R^2$ value is only 0.3.

Table 4

Parameter values and $R^2$ for fits of the model in (22) to the ten simulation model runs

Trial	a	g	$\gamma $	r	$\zeta $	b	$R^2$
1	0	0	0.00639	2.57354	0.135262	9.3676	0.98
2	5.10245	5.03148	7.72966	8.39135	0.041283	0	0.94
3	2.6741	2.6143	3.07877	4.29054	0.0112	0.0091	0.94
4	2.29737	2.26315	5.4087	7.28195	0.0916	0.0328	0.91
5	3.25968	3.21491	8.00409	7.97228	0.0482	0	0.91
6	1.49433	1.37595	4.89054	7.44912	0.0991	0.0224	0.62
7	0.207559	0.439949	0	1.17589	0.00969	7.89728	0.87
8	0.370378	0.280655	0.388098	18.4207	1.38813	0.326602	0.30
9	2.38719	2.8625	1.66452	10.1071	3.85e-006	2.53969	0.80
10	0.395282	0.366711	0	1.95892	0.00948	1.42142	0.93

Fig. 9
Stability analysis for Trial 8 fit with Type II functional response for a range of values of grazing rate (g) and time between hurricanes (K). Aside from g and K, all other parameters are from Trial 8 in Table 4. The *blue line* represents pairs (g, K) for which the cycle is unstable. Pairs above and to the *right* of the *blue line* are consistent with stable cycles. No cycles were detected for points below and to the *left* of the *blue line* (Color figure online)

Birfurcation

In Fig. 7, we considered bifurcations about parameter values from Trial 8 with a Type I functional response. If we consider the fits for Trial 8 as in Table 4 with the Type II functional response, then we have the bifurcation diagram as below in Fig. 9. What we notice is that for the parameters chosen the model with the Type II functional response has a much more forgiving bifurcation diagram. In other words, the grazing rate must become very low before hurricane frequency becomes an issue. To see if this is generally the case for Trial 9 we accordingly compared the Type I and Type II functional responses in Figs. 10 and 11. Here, however, we see that the model with the Type I functional response is more forgiving. Overall, the range of dynamics for models with either response function seems to be similar.

Fig. 10
Stability analysis for Trial 9 fit with Type I functional response for a range of values of grazing rate (g) and time between hurricanes (K). Aside from g and K, all other parameters are from Trial 9 in Table 4. The *blue line* represents pairs (g, K) for which the cycle is unstable. Pairs above and to the *right* of the *blue line* are consistent with stable cycles. No cycles were detected for points below and to the *left* of the *blue line* (Color figure online)

Fig. 11
Stability analysis for Trial 9 fit with Type II functional response for a range of values of grazing rate (g) and time between hurricanes (K). Aside from g and K, all other parameters are from Trial 9 in Table 4. The *blue line* represents pairs (g, K) for which the cycle is unstable. Pairs above and to the *right* of the *blue line* are consistent with stable cycles. No cycles were detected for points below and to the *left* of the *blue line* (Color figure online)

Model Selection

The model with the Type II functional response is generally preferred as being derived from first principles and experimentally. Using the Type II response allows a slightly better fit of the trial data compared with using the Type I functional response as noted comparing Tables 3 and 4. This, however, does not imply we should use this model to describe the data. What also needs to be considered here is the principle of parsimony. Biologically, accuracy is extremely important; however, we must be careful about using parameters that the data do not provide enough information to represent. Here the source of data is a simulation where grazing is best represented by a Type I response. If we insist on using the Type II, then our lack of knowledge about the value of the parameter b will likely lead to overfitting the model. Consequently, instead of introducing more biological accuracy, the addition of the parameter b can result in more bias for the system. A good discussion of this point be found on pages 132–134 in Burnham and Anderson (2002).

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Authors and Affiliations

Stephen Ippolito
- 1
Email author
Vincent Naudot
- 1
Erik G. Noonburg
- 2

1.Department of Mathematical SciencesFlorida Atlantic UniversityBoca RatonUSA
2.Department of Biological SciencesFlorida Atlantic UniversityDavieUSA

Alternative Stable States, Coral Reefs, and Smooth Dynamics with a Kick

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