The role of fear in a time-variant prey–predator model with multiple delays and alternative food source to predator

Abstract

We explore the dynamics of a time-variant prey–predator model of two species of fishes, namely Tenualosa ilisha (prey) and Macrognathus pancalus (predator). The prey exhibits anti-predatory behavior due to fear of predation. The prey equation is also equipped with age-based growth and harvestation terms to investigate the significance of maturity and harvest delay. The prey moves from saline water to freshwater for spawning, during which the predator feeds on an alternative food source. We explore the impact of this food source on the survival of prey. Starting with the sufficient conditions for positivity and boundedness of the solutions to the model, we find conditions for the existence of a stable global solution, periodic and almost periodic solutions. We find the range of existence for prey and predator populations and observe the interdependence between these values. According to our findings, just the fear of predation is enough to drive the prey population low. Even when the prey happens to be in an advantageous scenario, fear can still bring its population down, unless there are no or very few predators.

Appendix

The effect of fear is incorporated at the very basic step of the model. In order to understand the upcoming explanation, we have to keep in mind that it is a scientific modification of the logistic growth model. We know that the logistic growth model is given by

$$\begin{aligned}&\hbox {Rate of change of the population} \\&\quad = \hbox {Natural/linear birth/reproduction }\\&\quad - \hbox {Natural/linear death} \\&\quad - \hbox {death due to competition among themselves.} \end{aligned}$$

When written mathematically,

$$\begin{aligned} N' (t) = b(t)N(t)- \delta (t)N(t)-c(t) N^2 (t), \end{aligned}$$

(A1)

where N(t) is the density of the population (for the present system it is a prey population) at time $t, b(t), \delta (t)$, and c(t) are natural/linear birth/reproduction rate, natural/linear death rate and death rate due to competition among themselves for resources. But when predator (P(t)) is incorporated into the system, then due to fear, natural/linear birth/reproduction rate (b(t)) is affected (refer to [7]) and, it becomes b(t)f(k(t), P(t)) where $f(k(t),P(t))=\dfrac{1}{(1+k(t)P(t))}$ (fear-effect term). Finally, the logistic model becomes

$$\begin{aligned} N' (t) = \dfrac{b(t)}{(1+k(t)P(t))} N(t)- \delta (t) N(t)- c(t) N^2 (t),\nonumber \\ \end{aligned}$$

(A2)

where $\dfrac{b(t)}{1+k(t)P(t)}$ indicates the modified birth rate due to fear of predator.

Now according to the characteristics of prey (in the present system, it is a sexually reproductive species), it can give birth/reproduce after becoming sexually mature. That means those born now, they cannot give birth now, but those born before $\tau _1$ time from now and still alive and manage to survive during $t- \tau _1$ time which has attained maturity (can reproduce) and contributes to the birth rate of the population. To obtain an equation that describes how many individuals alive at time $t- \tau _1$ are still alive at time t, we solve the following rst order ODE for N(t) as a function of $N(t- \tau _1)$, where $N(t- \tau _1 )$ is the density of prey at time $t- \tau _1$. $\{- \delta (t) N(t)-c(t) N^2 (t) \}$ is the total death at time t, so:

$$\begin{aligned} \dfrac{dN(t)}{dt} = - \delta (t) N(t)- c(t) N^2 (t). \end{aligned}$$

Using the technique of separation of variables and integrating on both the sides from $t- \tau _1$ to t, it follows that

$$\begin{aligned} \int _{N(t-\tau _1)}^{N(t)} \dfrac{1}{\delta (t)N(t)+c(t) N^2 (t)} dN = - \int _{t- \tau _1}^{t} dt. \end{aligned}$$

And hence,

$$\begin{aligned} N(t) = \dfrac{\delta (t) N(t-\tau _1)}{\delta (t) e^{\delta (t)\tau _1}+c(t)(e^{\delta (t)\tau _1}-1)N(t-\tau _1)} \end{aligned}$$

(A3)

is the population that contributes to the birth rate at time t in prey population. Now we replace it in equation (A2); thus, we have

$$\begin{aligned} N'(t)= & {} \dfrac{b(t)}{(1+k(t)P(t))} \nonumber \\&\times \bigg [\dfrac{\delta (t) N(t-\tau _1)}{\delta (t) e^{\delta (t)\tau _1}+c(t)(e^{\delta (t)\tau _1}-1)N(t-\tau _1)}\bigg ] \nonumber \\&- \delta (t) N(t)- c(t) N^2 (t). \end{aligned}$$

(A4)

So, finally the modified logistic equation is:

$$\begin{aligned} \begin{array}{c} \begin{aligned} N'(t) &{}= \dfrac{b(t) \delta (t) N(t-\tau _1)}{[\delta (t) e^{\delta (t)\tau _1}+c(t)(e^{\delta (t)\tau _1}-1)N(t-\tau _1)][1+k(t)P(t)]} \\ &{}- \delta (t) N(t)- c(t) N^2 (t).\nonumber \end{aligned} \end{array}\!\!\!\\ \end{aligned}$$

(A5)

In the present study, the hilsa fish population (if we think about a single population model in the absence of the predator) is given by (along with the harvesting term, which can be derived in a similar manner, i.e., those born before $\tau _2$ time from now and alive during $t- \tau _2$ time, these are economically profitable and ecologically sustainable, we can harvest them):

$$\begin{aligned} \begin{array}{c} \begin{aligned} N'(t) &{}= \dfrac{b(t)\delta (t) N(t-\tau _1)}{\delta (t) e^{\delta (t)\tau _1}+c(t)(e^{\delta (t)\tau _1}-1)N(t-\tau _1)} \\ &{}\quad -\delta (t) N(t)- c(t) N^2(t)\\ &{}\quad -\dfrac{q(t)E(t)\delta (t)N(t-\tau _2)}{\delta (t) e^{\delta (t)\tau _2}+c(t)(e^{\delta (t)\tau _2}-1)N(t-\tau _2)}, \end{aligned} \end{array} \end{aligned}$$

(A6)

where q(t), and E(t) are catchability coefficient and efforts put into harvesting prey, respectively; $\tau _1$, and $\tau _2$ are maturation delays for reproduction and harvesting, respectively.

Next, we consider a predator to the model, then due to the foraging tendency of the predator (towards its prey, for its food), one direct effect (predation, here we consider Beddington-DeAngelis functional response with an alternative food source to the predator) and one fear-effect term (in prey) will be incorporated into the model. Thus, the final prey–predator model (i.e., equation (1.1)) is as follows:

$$\begin{aligned} \begin{array}{c} \begin{aligned} N'(t) &{}=\dfrac{b(t)\delta (t) N(t-\tau _1)}{[\delta (t) e^{\delta (t)\tau _1}+c(t)(e^{\delta (t)\tau _1}-1)N(t-\tau _1)][1+k(t)P(t)]}\\ &{}\quad -\delta (t) N(t)-c(t)N^2(t)\\ &{}\quad - \dfrac{\alpha (t)p(t)N(t)P(t)}{a(t)+p(t)N(t)+(1-p(t))A(t)+\xi (t)P(t)}\\ &{}\quad -\dfrac{q(t)E(t)\delta (t)N(t-\tau _2)}{\delta (t) e^{\delta (t)\tau _2}+c(t)(e^{\delta (t)\tau _2}-1)N(t-\tau _2)},\\ P'(t) &{}= \dfrac{\beta (t) \{p(t)N(t)+(1-p(t))A(t)\}P(t)}{a(t)+p(t)N(t)+(1-p(t))A(t)+\xi (t)P(t)}\\ &{}\quad -m(t)P(t). \end{aligned} \end{array} \end{aligned}$$