Dynamics of Planar Systems That Model Stage-Structured Populations

We study a general discrete planar system for modeling stage-structured populations. Our results include conditions for the global convergence of orbits to zero (extinction) when the parameters (vital rates) are time and density dependent. When the parameters are periodic we obtain weaker conditions for extinction. We also study a rational special case of the system for Beverton-Holt type interactions and show that the persistence equilibrium (in the positive quadrant) may be globally attracting even in the presence of interstage competition. However, we determine that with a sufficiently high level of competition, the persistence equilibrium becomes unstable (a saddle point) and the system exhibits period two oscillations.


Introduction
Stage-structured models of single-species populations with lowest dimension in discrete time are expressed as planar systems of difference equations.For a general expression of these models, consider system  ( + 1) =  1 ()  1 ( 11 ()  () ,  12 ()  ())  () +  2 ()  2 ( 21 ()  () ,  22 ()  ())  () , ( + 1) =  ()  ( 1 ()  () ,  2 ()  ())  () (1b) from [1] in which () and () are numbers (or densities) of juveniles and adults, respectively, remaining after  (juvenile) periods.The vital rates   and  (survival and inherent low density fertility) as well as the competition coefficients   and   in (1a) and (1b) may be density dependent; that is, they may depend on  and  and also explicitly on time; that is, the system may be nonautonomous.Early examples of matrix models used in species populations dynamics can be found in [2][3][4][5] and their comprehensive treatment is provided in [6].Under certain constraints on the various functions, including periodic vital rates and competition coefficients having the same common period , sufficient conditions for global convergence to zero (extinction) as well as the existence of periodic orbits for (1a) and (1b) are established in [1].If  is the mean fertility rate (the mean value of () above), then it is also shown that orbits of period  appear when  exceeds a critical value   , while global convergence to 0 or extinction occurs when  <   .On the other hand, conditions under which the species survives (i.e.permanence) were studied in [7,8].
In this paper, we study the following abstraction of the matrix model (1a) and (1b): where for each time period  ≥ 0 the functions  1, ,  2, ,   : [0, ∞) 2 → [0,∞) are bounded on the compact sets in [0, ∞) 2 .This feature allows for (0, 0) to be a fixed point of the system and it is true if, for example,  1, ,  2, ,   are continuous functions for every .Under biological constraints on the parameters, we may think of   = () and   = () as in (1a) and (1b).System (2a) and (2b) includes typical stage-structured models in the literature.For instance, the tadpole-adult model for the green tree frog Hyla cinerea population that is proposed in [9] may be expressed as This is a system of type (2a) and (2b) with Beverton-Holt type functions  1 and  2 .Competition in (3a) and (3b) occurs separately among juveniles and adults but not between the two classes, as they feed on separate resources; thus  1 and  2 do not depend on both juvenile and adult numbers and  is independent of both numbers.Two cases are analyzed in [9]: (i) continuous breeding with constant   =  so that (3a) and (3b) is autonomous and (ii) seasonal breeding where   is periodic.In addition to considering extinction and survival in the autonomous case, it is shown that seasonal breeding may be deleterious (relative to continuous breeding) for populations with high birth rates, but it can be beneficial with low birth rates.
Another system of type (2a) and (2b) is the autonomous stage-structured model with harvesting that is discussed in [10,11], which may be written as The numbers ℎ  , ℎ  ∈ [0,1] denote the harvest rates of juveniles and adults, respectively.The stock-recruitment function  : [0, ∞) → [0, ∞) may be compensatory (e.g., Beverton-Holt [12]) or overcompensatory (e.g., Ricker [13]).Compensatory recruitment is used in populations where recruitment increases with increase in densities before reaching an asymptote, while in overcompensatory models recruitment declines as density increases (see [11,14]).A thorough analysis of the dynamics of (4a) and (4b) with the Ricker function appears in [10].The results in [10,11] clarify many issues with regard to the effects of harvesting in stage-structured models such as global convergence to 0 and the existence of a stable survival equilibrium as well as the so-called hydra effect for different harvesting scenarios and with different recruitment functions; this refers to the counter-intuitive situation where an increase in the harvest or mortality rate results in a corresponding increase in the total population; for example, see [15][16][17].
Also studied in [10] is the occurrence of periodic and nonperiodic attractors and chaotic behavior for certain parameter ranges.
Next, the model in [18] studies the harvesting and predation of sex-and age-structured populations.Although the added stage for two sexes results in a three-dimensional model, the existence of an attracting, invariant planar manifold reduces the study of the asymptotics of the system to that of the planar system: where the density-dependent per capita reproductive rate  may be Beverton-Holt or Ricker similarly to  in (4b).Here   is the number of females and   is the number of young members in the population (the male population is a fixed proportion of the females).
We also mention the adult-juvenile model in which all adults are removed through harvesting, predation, migration, or just dying after one period, as in the case of semelparous species, that is, an organism that reproduces only once before death.In [19] conditions for the global attractivity of the positive fixed point and the occurrence of two cycles for (6a) and (6b) are obtained.A significant difference between (5a), (5b), (6a), and (6b) and systems (3a), (3b), (4a), and (4b) is the fact that  +1 in (5b) or in (6b) may depend on both   and   .
We study the qualitative properties of the orbits of (2a) and (2b) such as uniform boundedness and global convergence to 0 under minimal restrictions on time-dependent parameters.Biological constraints may be readily imposed to obtain special cases relevant to population models.
We also investigate convergence to zero with periodic parameters (extinction in a periodic environment).In particular, we show that convergence to zero occurs even if the mean value of  2, exceeds 1, a situation that cannot occur if  2, is constant in ; see Remark 16 below.
In the final section we study the dynamics of a rational special case of (2a) and (2b).Sufficient conditions for the global asymptotic stability of a fixed point in the positive quadrant [0, ∞) 2 as well as conditions for the occurrence of orbits of prime period two are obtained.In particular, we establish that a sufficiently high level of interspecies competition tends to destabilize the survival fixed point and result in periodic oscillations.

Global Attractivity of the Origin
In this section we obtain general sufficient conditions for the convergence of all orbits of the system to (0, 0).For population models these yield conditions that imply the extinction of species.
Throughout this section we assume that  , ,   are all bounded functions for  = 1, 2 and every  = 0, 1, 2, . ... Then the following are well-defined sequences of real numbers: (, V) .
Proof.By (26) there is  ∈ (0, 1) such that  1,  −1 +  2, ≤  for all (large) .From (2a) so for all (large)  (2b) yields Lemma 4 now implies that lim  → ∞   = 0. Furthermore either by hypothesis there is a positive number  such that   ≤  or by (27) there is a positive number  such that  1, ≥  for all (large)  so that for all sufficiently large values of .Now, if  =  or  = / as the case may be, then from (2b) in the planar system we see that lim and the proof is complete.
(2) If ( 26) is satisfied but {  } is unbounded and { 1, } does not satisfy (27) then   may not converge to 0; see the example following Corollary 18 below.
We consider an application of Theorem 5 to "noisy" autonomous system next.Let   ,  , ,  = 1, 2, be bounded sequences of real numbers and let Additionally, let  1 ,  2 ,  : [0, ∞) 2 → [0,∞) be bounded functions and denote their supremums over [0, ∞) 2 by  1 ,  2 , , respectively.If in (2a) and (2b) we have then we refer to (2a) and (2b) as an autonomous system with low-amplitude disturbances or fluctuations in the rates  1 ,  2 , , assuming that all three of these are positive functions and for all , V ≥ 0 These inequalities ensure that the functions   and  , are positive, as required for (2a) and (2b).
Corollary 7. Suppose that (2a) and ( 2b) is an autonomous system with low-amplitude disturbances or fluctuations in the above sense.If then the origin is the unique, globally asymptotically stable fixed point of ( 2a) and (2b) relative to the positive quadrant [0, ∞).
Remark 8. Since in the above discussion the sequences   ,  , ,  = 1,2, are arbitrary bounded sequences, they can also be sequences of random variables that are drawn from distributions with finite support.For example,   ,  , can be drawn from uniform distribution on some interval [0, ] so long as Corollary 7 will hold, implying that the origin is globally attracting even in the presence of noise.
In the autonomous case where the three parameter functions  1, ,  2, ,   do not depend on  at all, we have the following planar system: If in Corollary 7 we set   ,  = 0 in (35) then we obtain the following result for the above autonomous system.

functions and the following inequality holds:
then the origin is the unique, globally asymptotically stable fixed point of (37a) and (37b) relative to the positive quadrant [0, ∞) 2 .
Inequality (38) may be explicitly related to the local asymptotic stability of the origin for (37a) and (37b) when the functions  1 ,  2 ,  are smooth.Consider the associated mapping whose linearization at (0, 0) has eigenvalues These are real and a routine calculation shows that Under suitable differentiability hypotheses, this inequality is implied by (38) and is equivalent to it if the suprema of  2 and  1  occur at (0, 0).
Remark 10.An even simpler reduction than the above is possible if   (, V) =   () is independent of (or constant in) V.In this case, and it is not necessary to solve (2a) for   implicitly (i.e., the system folds without inversions).Special cases of this type include systems (3a), (3b), (4a), and (4b).

Global Convergence to Zero with Periodic Parameters.
The results in this section show that global convergence to zero may occur even if (26) does not hold; see Remark 16 below.Recall from the proof of Theorem 5 that The right-hand side of the above inequality is a linear expression.Consider the linear difference equation where the coefficients   and   are nonnegative and their periods  1 and  2 are positive integers with least common multiple  = lcm( 1 ,  2 ); we say that the linear difference equation ( 48) is periodic with period .In this study we assume that By Lemma 4 every solution of (48) converges to zero if   +   < 1 for all .However, it is known that convergence to zero may occur even when   +   exceeds 1 (for infinitely many  in the periodic case).We use the approach in [35] to examine the consequences of this issue when the planar system has periodic parameters.The following result is an immediate consequence of Theorem 13 in [35].
Lemma 11.Assume that   ,   for  = 1, 2, . . .,  are obtained by iteration from (48) from the real initial values: Suppose that the quadratic polynomial is proper, that is, not 0 = 0, and suppose that it has a real root generates nonzero real numbers  2 , . . .,   then {  } ∞ =1 is periodic with preiod  and yields a triangular system of firstorder equations that is equivalent to (48) as follows: System ( 53) and ( 54) is also known as a semiconjugate factorization of (48); see [36] for an introduction to this concept.The sequence {  } that is generated by ( 52) is said to be (unitary) eigensequence of (48).Eigenvalues are essentially constant eigensequences for if  = 1 in Lemma 11 then (51) reduces to and the latter equation is the standard characteristic equation of (48) with constant coefficients; see [35] for more details on the semiconjugate factorization of linear difference equations.
Each of ( 53) and ( 54) readily yields a solution by iteration as follows: Proof.(a) Let  1 = 0. Then  2 =  1 = 0 and since  1 = 0 by definition it follows that  3 = 0. Induction completes the proof that   = 0 if  ≥ 2. The converse is obvious since Now, the proof is completed by induction.The proof of (59) is similar since and if (59) holds for some  ≥ 2 then which establishes the induction step.
(b) If   > 0 for  = 1, . . .,  then Proof.(a) Lemma 12 shows that   > 0 for  = 2, . . .,  + 1.Now, either (i)  1 > 0 or (ii)  1 = 0.In case (i), the root  + of the quadratic polynomial (51) is positive since by Lemma 12  +1 > 0 and thus (b) To establish (63), let  1 =  + and note that (51) can be written as Since {  } has period ,  +1 =  1 so from (52) and the definition of the numbers   and   it follows that Since   ̸ = 0 it follows that We claim that if   ̸ = 0 for  = 1, . . .,  then This claim is easily seen to be true by induction; we showed that it is true for  = 0 and if (70) holds for some  then by ( 52) from which it follows that and the induction argument is complete.Now, using (70), we obtain Given that  1 =  + (73) implies that and ( 63) is obtained.Hence, Upon rearranging terms and squaring, which reduces to (64) after straightforward algebraic manipulations.
(c) First, assume that  is odd.Then by ( 59) so from ( 63) as required.Now let  be even.Then from ( 63) and ( 59) If   < 1 for  = 1, . . .,  then  2  4 ⋅ ⋅ ⋅   ≥  1  2 ⋅ ⋅ ⋅   and  1  3 ⋅ ⋅ ⋅  −1 ≥  1  2 ⋅ ⋅ ⋅   and the proof is complete.Some of the numbers   may exceed 1 in Lemma 13 without affecting the conclusions of the lemma.Additionally, not all the conditions in Lemma 13 are necessary.For instance, if  1 = 0 then Lemma 13(c) holds trivially.Additionally, by Lemma 12(a),   = 0 for  ≥ 2 so the following equality must hold instead of (63): This is in fact true because  1 =  + =  +1 /  so repeating the argument in the proof of Lemma 13(b) yields as claimed.These observations establish the following version of Lemma 13.

Remark 16.
(1) Condition (64) involves the numbers   ,   rather than the coefficients of (48) directly.In the case of period  = 2 the role of   and   is more apparent.Inequality (64) in this case is and simple manipulations reduce the last inequality to (2) Inequality (85) holds even if  1 > 1 or  2 > 1 thus showing how global convergence to (0, 0) my occur when (26) does not hold.Furthermore, it is possible that (85) holds together with Note that (85) holds even with arbitrarily large mean value in (86) if say  1 → 0 as  2 → ∞.In population models this implies that if (85) holds with   =  2, and   =  1,  −1 then extinction may still occur after restocking the adult population to raise the mean value of the composite parameter  2, above 1 by a wide margin.
(3) In Theorem 15 the individual sequences  1, ,   need not be periodic or even bounded.Therefore, the theorem applies to (2a) and (2b) even if the system itself is not periodic as long as the combination  1,  −1 of parameters is periodic along with  2, .

Dynamics of a Beverton-Holt Type Rational System
In this section we apply some of the preceding results and obtain some new ones to study boundedness, extinction, and modes of survival in some rational special cases of (2a) and (2b).In population models these types of systems include the Beverton-Holt type interactions.Specifically, we consider the following nonautonomous system and some of its special cases: where we assume that for all  ≥ 0 and  = 1, 2  1, > 0,   ,  2, ,  , ,  , ,  , ≥ 0   > 0 for infinitely many . ( For example, if we think of   as the natural survival rates then the population model (3a) and (3b) is a special case of (87a) and (87b).If we allow   to include additional factors such as harvesting rates then (87a) and ( 87b) is an extension of the model in [11] (with a Beverton-Holt recruitment function) in the sense that the competition coefficients  , ,  , , and  , may be nonzero as well as time-dependent.

Uniform Boundedness and Extinction.
We now examine boundedness and global convergence to 0 (extinction) in (87a) and (87b).The next result is in part a consequence of Corollary 3.
(b) Let the sequence {  } be bounded and suppose that there is  > 0 such that for all  larger than a given positive integer.Then all orbits of (87a) and (87b) are uniformly bounded.
Proof.(a) By hypothesis, for all (large) , Next, let By hypothesis, there is  1 > 0 and  ∈ (0, 1) such that for all , V ≥ 0 and all sufficiently large values of Now an application of Corollary 3 completes the proof of (a).
(b) By ( 89) for all large  it follows that and, likewise, for all large .Therefore,   ≤ 2.Next, if {  } is bounded then   ≤ 2  is also bounded and the proof is complete.
The next result follows readily from Theorem 5.
The above corollary is false when (95) holds if   is unbounded and thus  1, has a subsequence that converges to 0.
Corollary 17 takes a simpler form for the autonomous special case of (87a) and (87b); namely, with constant parameters The following result is applicable to (3a) and (3b) as well as special cases of (4a), (4b), (5a), and (5b) with rational .
Corollary 20.Assume that (99) holds.All orbits of (98a) and (98b) in [0, ∞) 2 are uniformly bounded if either one of the following conditions holds: It is noteworthy that if in part (a) above  1 = 0 then (98a) and (98b) may have unbounded solutions as in, for example, system where  2 =  1 = 0 and the remaining parameters are positive.This system folds to the second-order rational equation which is known to have unbounded solutions if  1  > 1; see [37].Corollary 18 likewise simplifies in the autonomous case.

Persistence and the Role of Competition.
We now explore the effects of competition in the autonomous system (98a) and (98b).There are 6 different competition coefficients and to reduce the number of different cases we focus on the special case below where   ,   = 0: If   define the natural survival rates   , then this system is complementary to (3a), (3b), (4a), and (4b) in the sense that in both of those systems  2 = 0.
By the last two corollaries, all orbits of the rational system (102) and ( 103) in [0, ∞) 2 are uniformly bounded if  1 > 0 and  2 < 1 and they converge to the origin if  1  +  2 < 1.We now examine this rational system in more detail using its folding, namely, the second-order rational equation where See (45); -component is given by (44) or calculated directly using (102) as With initial values  0 and  1 =  1  0 +  2  0 derived from ( 0 ,  0 ) ∈ [0, ∞) 2 , -component of the orbits {(  ,   )} of the system is obtained by iterating (104).The equation in (106) is passive in the sense that after -component of the orbit is generated by the core equation ( 104) -component is derived from (106) without any further iterations.This observation also establishes the nontrivial fact that solutions of (104) that correspond to the orbits of the system in [0, ∞) 2 are nonnegative and well-defined even for  < 0.
If  1  +  2 < 1, that is,  < 1 − , then zero is the only fixed point of (104).Corollary 21 establishes that, in this case, zero is globally asymptotically stable relative to [0, ∞).On the other hand, when  1  +  2 > 1, that is,  > 1 − , then 0 is no longer a stable fixed point of (104).By routine calculations, one can show that zero is a saddle point when 1− <  < 1+ and if  > 1 +  then zero is a repeller.
In addition, when  > 1 −  and  =  2 < 1, system (102) and (103) also has a fixed point in (0, ∞) 2 given by We note that  is also a positive fixed point of folding (104).Under certain conditions,  attracts all solutions of (104) with positive initial values, and it is thus a survival equilibrium.We state the following result from literature; see [38].( Then  is an invariant interval of ( 108 The rest of the proof follows from Lemma 22.

Proof. Since
to ensure that   ≥ 0 it suffices for 2 −  ≥ 0; that is, which is equivalent to and the proof is complete.
If  2 is sufficiently large then   is not positive on (0, ∞).Furthermore,  also becomes unstable for large enough  2 , which we establish next by examining the linearization of (104) around .
The characteristic equation associated with the linearization of (104) at  is given by where The roots of (119) are given by Since  V (, V) > 0 for all , V ∈ (0, ∞) it follows that  > 0 and both roots are real with  1 < 0 and  2 > 0. Furthermore,  2 < 1 if which is equivalent to This inequality holds, since  > 0 under our assumptions on the parameters.Therefore,  2 < 1.On the other hand,  1 > −1 if and only if which is equivalent to Note that when (1 − ) <  < (1 + ) this is trivially the case since  > 0 under our assumptions on the parameters.Thus,  is locally asymptotically stable if  < 1 + . Next, Then  ∈ (0, 1) Since (129) is equivalent to From the above inequality we obtain Thus if  2 >  then  is a saddle point and in particular the fixed point (, ) is unstable.These observations lead to the following which may be compared with Theorem 24.
Our final result establishes that when  2 > 0 is sufficiently large system (102) and (103) can have a prime period two orbit which occurs as  becomes unstable.Existence of periodic orbits is established via the folding in (104).
Simplifying the right hand side, it follows that Now, since we are assuming that  − (1 + ) > 0, then  − 1 > 0, so the right side of (143) is positive, which implies that  −  > 0. Stated differently, if  < , then (104) has no positive prime period two solution.
From (143) we get where ,  > 0 and To ensure that  and  are real, the roots of () must be real, which is the case if and only if  2 − 4 > 0; that is, The next result shows that a solution of period two appears when  loses its stability.from which we infer that  −  > 0 and the roots of () are guaranteed to be real and positive.This satisfies all the conditions of Theorem 27 which completes the proof.
Figure 1 shows two orbits of system (102) and (103) from initial points ( 0 ,  0 ) = (2.3, 1) and ( 0 ,  0 ) = (0.0001, 0.0001).Although both orbits converge to the period two cycle, a shadow of the stable manifold of the fixed point is also seen in the initial segments of the two orbits.If the initial points start exactly on the stable manifold of  then the solutions converge to .
We studied the dynamics of a general planar system that includes many common stage-structured population models that evolve in discrete time.Our hypotheses regarding system (2a) and (2b) and its parameters are more general than what is typically assumed in population models with the aim of gaining a broader understanding of the mathematical properties of the system.The study in this paper is rigorous but incomplete and many issues remain.Generalizing the results in Section 4 to a level closer to that in Section 3 leads to a more comprehensive treatment of planar or twostage, discrete population models.Among other things, this involves a consideration of systems involving the Ricker function where it is necessary to add the possibility of complex behavior.A resolution of these and related issues is left to future studies of system (2a) and (2b).

Corollary 28 .Figure 1 :
Figure 1: Orbits illustrating period two oscillations and the saddle point.